The matrix P = [ 0 0 4 0 4 0 4 0 0 ] is a (A) Square matrix (B) Diagonal matrix (C) Unit matrix (D) None

This is an Objective Type Questions as classified in NCERT Exemplar G i v e n t h a t A = [ 0 0 4 0 4 0 4 0 0 ] H e r e , n u m b e r o f c o l u m n s a n d t h e n u m b e r o f r o w s a r e e q u a l i . e . , 3 . S o , A i s a s q u a r e m a t r i x . H e n c e , t h e c o r r e c t o p t i o n i s ( a ) .

Total number of possible matrices of order 3 × 3 with each entry 2 or 0 is (A) 9 (B) 27 (C) 81 (D) 512

This is an Objective Type Questions as classified in NCERT Exemplar T o t a l n u m b e r o f p o s s i b l e m a t r i c e s o f o r d e r 3 × 3 w i t h e a c h e n t r y 0 o r 2 = 2 3 × 3 = 2 9 = 5 1 2 . H e n c e , t h e c o r r e c t o p t i o n i s ( d ) .

If [ 2 x + y 4 x 5 x − 7 4 x ] = [ 7 7 y − 1 3 y x + 6 ] , then the value of x + y is (A) x = 3 , y = 1 (B) x = 2 , y = 3 (C) x = 2 , y = 4 (D) x = 3 , y = 3

This is an Objective Type Questions as classified in NCERT Exemplar G i v e n t h a t : [ 2 x + y 4 x 5 x − 7 4 x ] = [ 7 7 y − 1 3 y x + 6 ] E q u a t i n g t h e c o r r e s p o n d i n g e l e m e n t s , w e g e t , 2 x + y = 7 … ( i ) a n d 4 x = 7 y − 1 3 … ( i i ) f r o m e q n . ( i i ) 4 x − x = 6 3 x = 6 ∴ x = 2 f r o m e q n . ( i ) 2 × 2 + y = 7 4 + y = 7 ∴ y = 7 − 4 = 3 H e n c e , t h e c o r r e c t o p t i o n i s ( b ) .

Find x , y , and z if: A=[02yzxy−zx−yz] satisfies AT=A−1 .

This is a Long Answers Type Questions as classified in NCERT Exemplar Sol.

Construct a2 × 2 matrix where (i) a i j = ( i − 2 j ) 2 2 . (ii) a i j = ? − 2 i + 3 j ? .

This is a short answer type question as classified in NCERT Exemplar L e t A = [ a 1 1 a 1 2 a 2 1 a 2 2 ] 2 × 2 ( i ) G i v e n t h a t a i j = ( i − 2 j ) 2 2 a 1 1 = ( 1 − 2 × 1 ) 2 2 = 1 2 ; a 1 2 = ( 1 − 2 × 2 ) 2 2 = 9 2 a 2 1 = ( 2 − 2 × 1 ) 2 2 = 0 ; a 2 2 = ( 2 − 2 × 2 ) 2 2 = 2 H e n c e , t h e m a t r i x A = [ 1 2 9 2 0 2 ] ( i i ) G i v e n t h a t a i j = | − 2 i + 3 j | a 1 1 = | − 2 × 1 + 3 × 1 | = 1 ; a 1 2 = | − 2 × 1 + 3 × 2 | = 4 a 2 1 = | − 2 × 2 + 3 × 1 | = − 1 ; a 2 2 = | − 2 × 2 + 3 × 2 | = 2 H e n c e , t h e m a t r i x A = [ 1 4 − 1 2 ]

Let x * y = x2 + y3 and (x * 1) * 1 = x * (1 * 1). Then a value of 2 s i n − 1 ( x 4 + x 2 − 2 x 4 + x 2 + 2 ) is.

(x2+1)*1=x*2 ⇒ (x2+1)2+1=x2+8⇒x2=2 2sin−1 (x4+x2−2x4+x2+2)=2sin−1 (12)=π3

Let R1 = { ( a , b ) ∈ N × N : | a − b | ≤ 1 3 } a n d R2 = { ( a , b ) ∈ N × N : | a − b | ≠ 1 3 } . Then on N :

R1 = { ( a , 1 ) ∈ N × N : | a − b | ≤ 1 3 } ( a , a ) ∈ R 1 a s | a − a | ≤ 1 3 ( R e f l e x i v e ) ( a , b ) & ( b , a ) ∈ R 1 a s | a − b | = | b − a | → ( s y m m e t r i c ) . But it is not necessary that if (a,b) & (b, c) ∈R, then (a,c)∈R Eg − ( 2 1 , 1 0 ) ∈ R & ( 1 0 , 1 ) ∈ R b u t ( 2 1 , 1 ) ∉ R 1 R2 = { ( a , b ) ∈ N × N : | a − b | ≠ 1 3 ∴ R 2 → N o t e q u i v a l e n c e s o l u t o i n . } ( a , b ) & ( b , a ) ∈ R 2 a s | a − b | = | b − a | But it is not necessary that if (a, b) & (b, c) ∈ R 2 then (a, c) also ∈ R 2 . Eg – (21, 1) ∈ R 2 & ( 1 , 8 ) ∈ R 2 b u t ( 2 1 , 8 ) ∉ R 2

The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives atleast 4 and atmost 7 candies, C3 receives atleast 2 and atmost 6 candies, is equal to:

Let the number of chocolates given to C1, C2, C3 & C4 be a, b, c, d respectively. Given 4 ≤ b ≤ 7 2 ≤ c ≤ 6 Now using these the maximum number of chocolates that can be given to C1 or C4 is 24 (where b & c are given 2 & 4 chocolates). ∴ 0 ≤ a ≤ 2 4 0 ≤ d ≤ 2 4 & a + b + c + d = 30 So, total possible solution to the above equation. Coefficient of x30 in. ( x 0 + x + x 2 + . . . . + x 2 4 ) ( x 4 + x 5 + . . . . + x 7 ) ( x 2 + x 3 + . . . . + x 6 ) ( x 0 + x + x 2 + . . . . + x 2 4 ) = (1+....+x24)2(x4)(1+x+....+x3)×x2(1+x+....+x4) = ( x 5 6 − 2 x 3 1 + x 6 ) × ( x 9 − x 4 − x 5 + 1 ) × ( x − 1 ) − 4 x56 & x31 can never give x30 so we discard them. x 6 × x 9 × ( x − 1 ) − 4 → 1 5 + 4 − 1 ? C 4 − 1 = 1 8 ? C 3 Coefficient x30 ® 18C3 – 23C3 – 22C3 + 27C3 = 1 8 × 1 7 × 1 6 6 − 2 3 × 2 2 × 2 1 6 − 2 2 × 2 1 × 2 0 6 + 2 7 × 2 6 × 2 5 6 = 430

Maths NCERT Exemplar Solutions Class 12th Chapter Three: Overview, Questions, Easy Tricks, Rules, Preparation

Q: ___ matrix is both symmetric and skew symmetric matrix.

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar N u l l m a t r i x i . e . [ 0 0 0 0 ] o r [ 0 0 0 0 0 0 0 0 0 ] i s b o t h s y m m e t r i c a n d s k e w s y m m e t r i c m a t r i x .

Q: Sum of two skew symmetric matrices is always ___ matrix.

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar L e t A a n d B b e a n y t w o m a t r i c e s ∴ F o r s k e w s y m m e t r i c m a t r i c e s A = − A ' … ( i ) a n d B = − B ' … ( i i ) A d d i n g ( i ) a n d ( i i ) w e g e t A + B = − A ' − B ' ⇒ A + B = − ( A ' + B ' ) , s o A + B i s s k e w s y m m e t r i c m a t r i x . H e n c e , t h e s u m o f t w o s k e w s y m m e t r i c m a t r i c e s i s a l w a y s s k e w s y m m e t r i c m a t r i c e s .

Q: The negative of a matrix is obtained by multiplying it by ___.

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar L e t A b e a m a t r i x ∴ − A = − 1 . A H e n c e , n e g a t i v e o f a m a t r i x i s o b t a i n e d b y m u l t i p l y i n g i t b y − 1 .

Matrices Long Answers Type Questions

Q1. If $A B = B A$ for any two square matrices, prove by mathematical induction that ${(A B)}^{n} = A^{n} B^{n}$ .

Sol.

$\begin{array}{l} L e t P (n) : {(A B)}^{n} = A^{n} B^{n} \\ S t e p 1 : P u t n = 1, P (1) : A B = A B w h i c h i s t r u e f o r n = 1 \\ S t e p 2 : P u t n = k, P (k) : {(A B)}^{k} = A^{k} B^{k} L e t i t b e t r u e f o r a n y k \in N \\ S t e p 3 : P u t n = k + 1, P (k + 1) : {(A B)}^{k + 1} = A^{k + 1} B^{k + 1} \\ L . H . S . {(A B)}^{k + 1} = {(A B)}^{k} . A B \\ = A^{k} B^{k} . A B [F r o m s t e p 2] \\ = A^{k + 1} A^{k + 1} R . H . S . \\ H e n c e, i f P (n) i s t r u e f o r P (k) t h e n i t i s t r u e f o r P (k + 1) . \end{array}$

Q2. Find $x$ , $y$ , and $z$ if: $A = [\begin{matrix} 0 & 2 y & z \\ x & y & - z \\ x & - y & z \end{matrix}]$ satisfies $A^{T} = A^{- 1}$ .

Sol.

Commonly asked questions

Q:

If $A B = B A$ for any two square matrices, prove by mathematical induction that ${(A B)}^{n} = A^{n} B^{n}$ .

Q:

Find $x$ , $y$ , and $z$ if: $A = [\begin{matrix} 0 & 2 y & z \\ x & y & - z \\ x & - y & z \end{matrix}]$ satisfies $A^{T} = A^{- 1}$ .

Q:

If possible, using elementary row transformations, find the inverse of the following matrices:

(i) $[\begin{matrix} 2 & - 1 & 3 \\ - 5 & 3 & 1 \\ - 3 & 2 & 3 \end{matrix}]$

(ii) $[\begin{matrix} 2 & 3 & - 3 \\ - 1 & - 2 & 2 \\ 1 & 1 & - 1 \end{matrix}]$

(iii) $[\begin{matrix} 2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{matrix}]$

Matrices Short Answers Type Questions

Q1. If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

Sol. $\begin{array}{l} T h e p o s s i b l e o r d e r s t h a t a m a t r i x h a v i n g 28 e l e m e n t s a r e {28 \times 1, 1 \times 28, 2 \times 14, 14 \times 2, 4 \times 7, 7 \times 4} . \\ T h e p o s s i b l e o r d e r s o f a m a t r i x h a v i n g 13 e l e m e n t s a r e {1 \times 13, 13 \times 1} . \end{array}$

Q2. In the matrix write:

(i) The order of the matrix .

(ii) The number of elements.

(iii) Write elements $a_{23}, a_{31}, a_{12}$ .

Sol.

$\begin{array}{l} (i) T h e o r d e r o f t h e g i v e n m a t r i x A i s 3 \times 3 \\ (i i) T h e n u m b e r o f e l e m e n t s i n a m a t r i x A = 3 \times 3 = 9 \\ (i i i) a_{i j} = t h e e l e m e n t s o f i^{t h} r o w a n d j^{t h} c o l u m n . \\ S o, a_{23} = x^{2} - y, a_{31} = 0, a_{12} = 1 . \end{array}$

Commonly asked questions

Q:

If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

Q:

In the matrix write:

(i) The order of the matrix .

(ii) The number of elements.

(iii) Write elements $a_{23}, a_{31}, a_{12}$ .

Q:

Construct a_{2 × 2} matrix where

(i) $a_{i j} = \frac{{(i - 2 j)}^{2}}{2}$ .
(ii) $a_{i j} = ? - 2 i + 3 j ?$ .

Matrices Objective Type Questions

Choose the correct answer from the given four options in each of the Exercises 1 to 15:

Q1. The matrix $P = [\begin{matrix} 0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0 \end{matrix}]$ is a

(A) square matrix

(B) diagonal matrix

(C) unit matrix

(D) none

Sol:

$\begin{array}{l} G i v e n t h a t A = [\begin{matrix} 0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0 \end{matrix}] \\ H e r e, n u m b e r o f c o l u m n s a n d t h e n u m b e r o f r o w s a r e e q u a l i . e ., 3 . S o, A i s a s q u a r e m a t r i x . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

Q2. Total number of possible matrices of order $3 \times 3$ with each entry 2 or 0 is

(A) 9

(B) 27

(C) 81

(D) 512

Sol:

$\begin{array}{l} T o t a l n u m b e r o f p o s s i b l e m a t r i c e s o f o r d e r 3 \times 3 w i t h e a c h e n t r y 0 o r 2 = 2^{3 \times 3} = 2^{9} = 512 . \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

Q3. If $[\begin{matrix} 2 x + y & 4 x \\ 5 x - 7 & 4 x \end{matrix}] = [\begin{matrix} 7 & 7 y - 13 \\ y & x + 6 \end{matrix}],$ then the value of $x + y$ is

(A) $x = 3, y = 1$

(B) $x = 2, y = 3$

(C) $x = 2, y = 4$

(D) $x = 3, y = 3$

Sol:

$\begin{array}{l} G i v e n t h a t : [\begin{matrix} 2 x + y & 4 x \\ 5 x - 7 & 4 x \end{matrix}] = [\begin{matrix} 7 & 7 y - 13 \\ y & x + 6 \end{matrix}] \\ E q u a t i n g t h e c o r r e s p o n d i n g e l e m e n t s, w e g e t, \\ 2 x + y = 7 \dots (i) \\ a n d 4 x = 7 y - 13 \dots (i i) \\ f r o m e q n . (i i) 4 x - x = 6 \\ 3 x = 6 \\ ∴ x = 2 \\ f r o m e q n . (i) 2 \times 2 + y = 7 \\ 4 + y = 7 ∴ y = 7 - 4 = 3 \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

Q4. If $A = \frac{1}{π} [\begin{matrix} s i n^{- 1} (x π) & t a n^{- 1} (\frac{x}{π}) \\ s i n^{- 1} (\frac{x}{π}) & c o t^{- 1} (π x) \end{matrix}], B = \frac{1}{π} [\begin{matrix} - c o s^{- 1} (x π) & t a n^{- 1} (\frac{x}{π}) \\ s i n^{- 1} (\frac{x}{π}) & - t a n^{- 1} (x π) \end{matrix}],$ then $A - B$ is equal to:

Sol:

$\begin{array}{l} G i v e n t h a t : A = \frac{1}{π} [\begin{matrix} s i n^{- 1} (x π) & t a n^{- 1} (\frac{x}{π}) \\ s i n^{- 1} (\frac{x}{π}) & c o t^{- 1} (π x) \end{matrix}] \\ a n d B = \frac{1}{π} [\begin{matrix} - c o s^{- 1} (x π) & t a n^{- 1} (\frac{x}{π}) \\ s i n^{- 1} (\frac{x}{π}) & - t a n^{- 1} (π x) \end{matrix}] \\ A - B = \frac{1}{π} [\begin{matrix} s i n^{- 1} (x π) & t a n^{- 1} (\frac{x}{π}) \\ s i n^{- 1} (\frac{x}{π}) & c o t^{- 1} (π x) \end{matrix}] - \frac{1}{π} [\begin{matrix} - c o s^{- 1} (x π) & t a n^{- 1} (\frac{x}{π}) \\ s i n^{- 1} (\frac{x}{π}) & - t a n^{- 1} (π x) \end{matrix}] \\ = \frac{1}{π} [\begin{matrix} s i n^{- 1} (x π) + c o s^{- 1} (x π) & t a n^{- 1} (\frac{x}{π}) - t a n^{- 1} (\frac{x}{π}) \\ s i n^{- 1} (\frac{x}{π}) - s i n^{- 1} (\frac{x}{π}) & c o t^{- 1} (π x) + t a n^{- 1} (π x) \end{matrix}] \\ = \frac{1}{π} [\begin{matrix} \frac{π}{2} & 0 \\ 0 & \frac{π}{2} \end{matrix}] [\begin{array}{l} ∵ s i n^{- 1} x + c o s^{- 1} x = \frac{π}{2} \\ t a n^{- 1} x + c o t^{- 1} x = \frac{π}{2} \end{array}] \\ = \frac{1}{π} \times \frac{π}{2} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = \frac{1}{2} I \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

Q5. If $A$ and $B$ are two matrices of the order $3 \times m$ and $3 \times n$ , respectively, and $m = n$ , then the order of the matrix $(5 A - 2 B)$ is

(A) $m \times 3$

(B) $3 \times 3$

(C) $m \times n$

(D) $3 \times n$

Sol:

$\begin{array}{l} A s w e k n o w t h a t t h e a d d i t i o n a n d s u b t r a c t i o n o f t w o m a t r i c e s i s o n l y p o s s i b l e w h e n t h e y h a v e \\ s a m e o r d e r . I t i s a l s o g i v e n t h a t m = n . \\ ∴ O r d e r o f (5 A - 2 B) i s 3 \times n . \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

Q6. If $A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ , then $A^{2}$ is equal to

(A) $[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

(B) $[\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}]$

(C) $[\begin{matrix} 0 & 1 \\ 0 & 1 \end{matrix}]$

(D) $[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Sol:

$\begin{array}{l} G i v e n t h a t A = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] \\ A^{2} = A . A = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] = [\begin{matrix} 0 + 1 & 0 + 0 \\ 0 + 0 & 1 + 0 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

Q7. If matrix $A = [a_{i j}]_{2 \times 2}$ , where

a_{i j} = 1

if $i \neq j$

if $i = j$ ,then $A^{2}$ is equal to

(A) I

(B) A

(C) 0

(D) None of these

Sol:

$\begin{array}{l} G i v e n t h a t A = {[a_{i j}]}_{2 \times 2} \\ L e t A = {[\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}]}_{2 \times 2} \\ a_{11} = 0 [∵ i = j] \\ a_{12} = 1 [∵ i \neq j] \\ a_{21} = 1 [∵ i \neq j] \\ a_{22} = 0 [∵ i = j] \\ ∴ A = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] \\ N o w, A^{2} = A . A = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] = [\begin{matrix} 0 + 1 & 0 + 0 \\ 0 + 0 & 1 + 0 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = 1 \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

Q8.The matrix $[\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{matrix}]$ is a

(A) Identity matrix

(B) Symmetric matrix

(C) Skew symmetric matrix

(D) None of these

Sol:

$\begin{array}{l} L e t A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{matrix}] \\ A^{'} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{matrix}] = A \\ A^{'} = A, s o A i s a s y m m e t r i c m a t r i x . \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

Q9. The matrix $[\begin{matrix} 0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0 \end{matrix}]$ is a

(A) Diagonal matrix

(B) Symmetric matrix

(C) Skew symmetric matrix

(D) Scalar matrix

Sol:

$\begin{array}{l} L e t A = [\begin{matrix} 0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0 \end{matrix}] \\ A^{'} = [\begin{matrix} 0 & 5 & - 8 \\ - 5 & 0 & - 12 \\ 8 & 12 & 0 \end{matrix}] \\ \Rightarrow A^{'} = - [\begin{matrix} 0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0 \end{matrix}] = - A \\ A^{'} = - A, s o A i s a s k e w s y m m e t r i c m a t r i x . \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

Q10. If $A$ is a matrix of order $m \times n$ and $B$ is a matrix such that $A B^{'}$ and $B^{'} A$ are both defined, then the order of matrix $B$ is

(A) $m \times m$

(B) $n \times n$

(C) $n \times m$

(D) $m \times n$

Sol:

$\begin{array}{l} O r d e r o f m a t r i x A = m \times n \\ L e t o r d e r o f m a t r i x B b e K \times P \\ I f A B^{'} i s d e f i n e d t h e n t h e o r d e r o f A B^{'} i s m \times K i f n = P \\ I f B^{'} A i s d e f i n e d t h e n t h e o r d e r o f B^{'} A i s P \times n w h e n K = m \\ N o w, o r d e r o f B^{'} = P \times K \\ ∴ o r d e r o f B^{'} = K \times P \\ = m \times n [∵ K = m, P = n] \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

Q11. If $A$ and $B$ are matrices of the same order, then $A B^{'} - B A^{'}$ is a

(A) Skew symmetric matrix

(B) Null matrix

(C) Symmetric matrix

(D) Unit matrix

Sol:

$\begin{array}{l} L e t P = (A B^{'} - B A^{'}) \\ P^{'} = {(A B^{'} - B A^{'})}^{'} \\ = {(A B^{'})}^{'} - {(B A^{'})}^{'} \\ = {(B^{'})}^{'} A^{'} - {(A^{'})}^{'} B^{'} [∵ {(A B)}^{'} = B^{'} A^{'}] \\ = B A^{'} - A B^{'} \\ = - (A B^{'} - B A^{'}) = - P \\ P^{'} = - P, s o i t i s a s k e w s y m m e t r i c m a t r i x . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

Q12. If $A$ is a square matrix such that $A^{2} = I$ , then ${(A - I)}^{3} + {(A + I)}^{3} - 7 A$ is equal to

(A) A

(B) I - A

(C) I + A

(D) 3A

Sol:

$\begin{array}{l} {(A - I)}^{3} + {(A + I)}^{3} - 7 A = A^{3} - I^{3} - 3 A^{2} I + 3 A I^{2} + A^{3} + I^{3} + 3 A^{2} I + 3 A I^{2} - 7 A \\ = 2 A^{3} + 6 A I^{2} - 7 A \\ = 2 A . A^{2} + 6 A I - 7 A \\ = 2 A I + 6 A I - 7 A [A^{2} = I] \\ = 8 A I - 7 A = 8 A - 7 A = A \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

Q13. For any two matrices $A$ and $B$ , we have

(A) AB = BA

(B) AB $AB \neq BA$ BA

(C) AB = O

(D) None of the above

Sol:

$\begin{array}{l} W e k n o w t h a t f o r a n y t w o m t r i c e s A a n d B, w e m a y h a v e A B = B A, A B \neq B A a n d A B = 0, b u t \\ i t i s n o t a l w a y s t r u e . \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

Q14. On using elementary column operations $C_{2} \to C_{2} - 2 C_{1}$ in the following matrix equation: $[\begin{matrix} 1 & - 3 \\ 2 & 4 \end{matrix}] = [\begin{matrix} 1 & - 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 3 & 1 \\ 2 & 4 \end{matrix}],$ we have:

(A) $[\begin{matrix} 1 & - 5 \\ 0 & 4 \end{matrix}] = [\begin{matrix} 1 & - 1 \\ - 2 & 2 \end{matrix}] [\begin{matrix} 3 & - 5 \\ 2 & 0 \end{matrix}]$

(B) $[\begin{matrix} 1 & - 5 \\ 0 & 4 \end{matrix}] = [\begin{matrix} 1 & - 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 3 & - 5 \\ 0 & 2 \end{matrix}]$

(C) $[\begin{matrix} 1 & - 5 \\ 2 & 0 \end{matrix}] = [\begin{matrix} 1 & - 3 \\ 0 & 1 \end{matrix}] [\begin{matrix} 3 & 1 \\ - 2 & 4 \end{matrix}]$

(D) $[\begin{matrix} 1 & - 5 \\ 2 & 0 \end{matrix}] = [\begin{matrix} 1 & - 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 3 & - 5 \\ 2 & 0 \end{matrix}]$

Sol:

$\begin{array}{l} G i v e n t h a t : [\begin{matrix} 1 & - 3 \\ 2 & 4 \end{matrix}] = [\begin{matrix} 1 & - 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 3 & 1 \\ 2 & 4 \end{matrix}] \\ Using C_{2} \to C_{2} - 2 C_{1}, w e g e t \\ [\begin{matrix} 1 & - 5 \\ 2 & 0 \end{matrix}] = [\begin{matrix} 1 & - 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 3 & - 5 \\ 2 & 0 \end{matrix}] \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

Q15. On using elementary row operation $R_{1} \to R_{1} - 3 R_{2}$ in the following matrix equation:

$[\begin{matrix} 4 & 2 \\ 3 & 3 \end{matrix}] = [\begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix}] [\begin{matrix} 2 & 0 \\ 1 & 1 \end{matrix}],$ we have:

(A) $[\begin{matrix} - 5 & - 7 \\ 3 & 3 \end{matrix}] = [\begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix}] [\begin{matrix} 2 & 0 \\ 1 & 1 \end{matrix}]$

(B) $[\begin{matrix} - 5 & - 7 \\ 3 & 3 \end{matrix}] = [\begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix}] [\begin{matrix} - 1 & - 3 \\ 1 & 1 \end{matrix}]$

(C) $[\begin{matrix} - 5 & - 7 \\ 3 & 3 \end{matrix}] = [\begin{matrix} 1 & 2 \\ 1 & - 7 \end{matrix}] [\begin{matrix} 2 & 0 \\ 1 & 7 \end{matrix}]$

(D) $[\begin{matrix} 4 & 2 \\ - 5 & - 7 \end{matrix}] = [\begin{matrix} 1 & 2 \\ - 3 & - 3 \end{matrix}] [\begin{matrix} 2 & 0 \\ 1 & 1 \end{matrix}]$

Sol:

$\begin{array}{l} W e h a v e, [\begin{matrix} 4 & 2 \\ 3 & 3 \end{matrix}] = [\begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix}] [\begin{matrix} 2 & 0 \\ 1 & 1 \end{matrix}] \\ Using e l e m e n t a r y r o w t r a n s f o r m a t i o n R_{1} \to R_{1} - 3 R_{2}, w e g e t \\ [\begin{matrix} - 5 & - 7 \\ 3 & 3 \end{matrix}] = [\begin{matrix} 1 & - 7 \\ 0 & 3 \end{matrix}] [\begin{matrix} 2 & 0 \\ 1 & 1 \end{matrix}] \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

Commonly asked questions

Q:

The matrix $P = [\begin{matrix} 0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0 \end{matrix}]$ is a

(A) Square matrix

(B) Diagonal matrix

(C) Unit matrix

(D) None

Q:

Total number of possible matrices of order $3 \times 3$ with each entry 2 or 0 is

(A) 9

(B) 27

(C) 81

(D) 512

Q:

If $[\begin{matrix} 2 x + y & 4 x \\ 5 x - 7 & 4 x \end{matrix}] = [\begin{matrix} 7 & 7 y - 13 \\ y & x + 6 \end{matrix}],$ then the value of $x + y$ is

(A) $x = 3, y = 1$

(B) $x = 2, y = 3$

(C) $x = 2, y = 4$

(D) $x = 3, y = 3$

Matrices Fill in the blanks Type Questions

Q1. ___ matrix is both symmetric and skew symmetric matrix.

Sol.

$N u l l m a t r i x i . e . [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] o r [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] i s b o t h s y m m e t r i c a n d s k e w s y m m e t r i c m a t r i x .$

Q2. Sum of two skew symmetric matrices is always ___ matrix.

Sol.

$\begin{array}{l} L e t A a n d B b e a n y t w o m a t r i c e s \\ ∴ F o r s k e w s y m m e t r i c m a t r i c e s \\ A = - A^{'} \dots (i) \\ a n d B = - B^{'} \dots (i i) \\ A d d i n g (i) a n d (i i) w e g e t \\ A + B = - A^{'} - B^{'} \\ \Rightarrow A + B = - (A^{'} + B^{'}), s o A + B i s s k e w s y m m e t r i c m a t r i x . \\ H e n c e, t h e s u m o f t w o s k e w s y m m e t r i c m a t r i c e s i s a l w a y s s k e w s y m m e t r i c m a t r i c e s . \end{array}$

Q3. The negative of a matrix is obtained by multiplying it by ___.

Sol.

$\begin{array}{l} L e t A b e a m a t r i x \\ ∴ - A = - 1 . A \\ H e n c e, n e g a t i v e o f a m a t r i x i s o b t a i n e d b y m u l t i p l y i n g i t b y - 1 . \end{array}$

Q4. The product of any matrix by the scalar ___ is the null matrix.

Sol.

$\begin{array}{l} L e t A b e a n y m a t r i x \\ ∴ 0 . A = A . 0 \\ H e n c e, t h e p r o d u c t o f a n y m a t r i x i s b y t h e r s c a l a r 0 i s t h e n u l l m a t r i x . \end{array}$

Q5. A matrix which is not a square matrix is called a ___ matrix.

Sol.

$A m a t r i x w h i c h i s n o t a s q u a r e m a t r i x i s c a l l e d a rectangular m a t r i x .$

Q6. Matrix multiplication is ___ over addition.

Sol:

$\begin{array}{l} M a t r i x m u l t i p l i c a t i o n i s d i s t r i b u t i v e o v e r a d d i t i o n . L e t A, B, a n d C b e a n y m a t r i c e s . \\ S o, (i) A (B + C) = A B + A C \\ (i i) (A + B) C = A C + B C \end{array}$

Q7. If $A$ is a symmetric matrix, then $A^{3}$ is a ___ matrix.

Sol.

$\begin{array}{l} L e t A b e a s y m m e t r i c m a t r i x \\ ∴ A^{'} = A \\ {(A^{3})}^{'} = {(A^{'})}^{3} = A^{3} [∵ {(A^{'})}^{k} = {(A^{k})}^{'}] \\ H e n c e, i f A i s a s y m m e t r i c m a t r i x, t h e n A^{3} i s a s y m m e t r i c m a t r i x . \end{array}$

Q8. If $A$ is a skew symmetric matrix, then $A^{2}$ is a ___.

Sol.

$\begin{array}{l} I f A i s a s k e w s y m m e t r i c m a t r i x, \\ ∴ A^{'} = - A \\ {(A^{2})}^{'} = {(A^{'})}^{2} = {(- A)}^{2} = A^{2} \\ H e n c e, A^{2} i s a s y m m e t r i c m a t r i x . \end{array}$

Q9. If $A$ and $B$ are square matrices of the same order, then

(i) ${(A B)}^{'} =$ ___.
(ii) ${(k A)}^{'} =$ ___ (k is any scalar).
(iii) ${[k (A - B)]}^{'} =$ ___.

Sol.

$\begin{array}{l} (i) {(A B)}^{'} = B^{'} A^{'} \\ (i i) {(k A)}^{'} = k . A^{'} \\ (i i i) {[k (A - B)]}^{'} = k {(A - B)}^{'} = k (A^{'} - B^{'}) \end{array}$

Q10. If $A$ is skew symmetric, then $k A$ is a ___ (k is any scalar).

Sol.

$\begin{array}{l} I f A i s a s k e w s y m m e t r i c m a t r i x \\ ∴ A^{'} = - A \\ {(k A)}^{'} = k A^{'} = k (- A) = - k A \\ H e n c e, k A i s a s k e w s y m m e t r i c m a t r i x . \end{array}$

Q11. If $A$ and $B$ are symmetric matrices, then

(i) $A B - B A$ is a ___.

(ii) $B A - 2 A B$ is a ___.

Sol.

$\begin{array}{l} (i) L e t P = (A B - B A) \\ P^{'} = {(A B - B A)}^{'} \\ = {(A B)}^{'} - {(B A)}^{'} \\ = B^{'} A^{'} - A^{'} B^{'} [∵ {(A B)}^{'} = B^{'} A^{'}] \\ = B A - A B [∵ A^{'} = A a n d B^{'} = B] \\ = - (A B - B A) = - P \\ H e n c e, (A B - B A) i s a s k e w s y m m e t r i c m a t r i x . \\ (i i) L e t Q = (B A - 2 A B) \\ Q^{'} = {(B A - 2 A B)}^{'} \\ = {(B A)}^{'} - {(2 A B)}^{'} \\ = A^{'} B^{'} - 2 {(A B)}^{'} [∵ {(k A)}^{'} = k A^{'}] \\ = A^{'} B^{'} - 2 B^{'} A^{'} \\ = A B - 2 B A [∵ A^{'} = A a n d B^{'} = B] \\ = - (2 B A - A B) \\ H e n c e, (B A - 2 A B) i s n e i t h e r a s y m m e t r i c m a t r i x n o r a s k e w s y m m e t r i c m a t r i x . \end{array}$

Q12. If $A$ is symmetric matrix, then $B^{'} A B$ is ___.

Sol.

$\begin{array}{l} I f A i s a s y m m e t r i c m a t r i x \\ ∴ A^{'} = - A \\ L e t P = B^{'} A B \\ P^{'} = {(B^{'} A B)}^{'} = B^{'} A^{'} {(B^{'})}^{'} [∵ {(A B)}^{'} = B^{'} A^{'}] \\ = B^{'} A B [∵ A^{'} = A a n d {(B^{'})}^{'} = B] \\ P^{'} = P \\ S o, P i s a s y m m e t r i c m a t r i x . \\ H e n c e, B^{'} A B i s a s y m m e t r i c m a t r i x . \end{array}$

Q13. If $A$ and $B$ are symmetric matrices of same order, then $A B$ is symmetric if and only if ___.

Sol.

$\begin{array}{l} (i) G i v e n t h a t A^{'} = A a n d B^{'} = B \\ L e t P = A B \\ P^{'} = {(A B)}^{'} \\ = B^{'} A^{'} \\ = B A [∵ A^{'} = A a n d B^{'} = B] \\ = P \\ H e n c e, A B i s s y m m e t r i c i f a n d o n l y i f A B = B A . \end{array}$

Q14. In applying one or more row operations while finding $A^{- 1}$ by elementary row operations, we obtain all zeros in one or more rows, then $A^{- 1}$ ___.

Sol.

$\begin{array}{l} A^{- 1} d o e s n o t e x i s t i f w e a p p l y o n e o r m o r e r o w o p e r a t i o n s w h i l e f i n d i n g A^{- 1} b y e l e m e n t a r y r o w \\ o p e r a t i o n s, o b t a i n a l l z e r o e s i n o n e o r m o r e r o w s . \end{array}$

Commonly asked questions

Q:

___ matrix is both symmetric and skew symmetric matrix.

Q:

Sum of two skew symmetric matrices is always ___ matrix.

Q:

The negative of a matrix is obtained by multiplying it by ___.

Matrices True or False Type Questions

Q1. A matrix denotes a number.

Sol.

$\begin{array}{l} F a l s e . \\ A m a t r i x i s a n a r r a y o f e l e m e n t s, n u m b e r s o r f u n c t i o n s h a v i n g r o w s a n d c o l u m n s . \end{array}$

Q2. Matrices of any order can be added.

Sol.

$\begin{array}{l} F a l s e . \\ T h e m a t r i c e s h a v i n g s a m e o r d e r c a n o n l y b e a d d e d . \end{array}$

Q3. Two matrices are equal if they have the same number of rows and the same number of columns.

Sol.

$\begin{array}{l} F a l s e . \\ T h e t w o m a t r i c e s a r e s a i d t o b e e q u a l i f t h e i r c o r r e s p o n d i n g e l e m e n t s a r e s a m e . \end{array}$

Q4. Matrices of different orders cannot be subtracted

Sol.

$\begin{array}{l} T r u e . \\ F o r a d d i t i o n a n d s u b t r a c t i o n, t h e o r d e r o f t w o m a t r i c e s s h o u l d b e s a m e . \end{array}$

Q5. Matrix addition is associative as well as commutative.

Sol.

$\begin{array}{l} T r u e . \\ I f A, B a n d C a r e t h e m a t r i c e s o f a d d i t i o n t h e n \\ A + (B + C) = (A + B) + C (a s s o c i a t i v e) \\ A + B = B + A (c o m m u t a t i v e) \end{array}$

Q6. Matrix multiplication is commutative.

Sol.

$\begin{array}{l} F a l s e . \\ Since A B \neq B A i f A B a n d B A a r e w e l l d e f i n e d . \end{array}$

Q7. A square matrix where every element is unity is called an identity matrix.

Sol.

$\begin{array}{l} F a l s e . \\ Since, i n a n i d e n t i t y m a t r i x a l l t h e e l e m e n t s o f p r i n c i p a l d i a g o n a l a r e u n i t y r e s t a r e z e r o . \\ e . g ., A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = I_{3} \end{array}$

Q8. If $A$ and $B$ are two square matrices of the same order, then $A + B = B + A$ .

Sol.

$\begin{array}{l} T r u e . \\ I f A a n d B a r e s q u a r e m a t r i c e s o f t h e n t h e i r a d d i t i o n i s c o m m u t a t i v e i . e ., A + B = B + A \end{array}$

Q9. If $A$ and $B$ are two matrices of the same order, then $A - B = B - A$ .

Sol.

$\begin{array}{l} F a l s e . \\ Since s u b t r a c t i o n s o f a n y t w o m a t r i c e s o f t h e s a m e o r d e r i s n o t c o m m u t a t i v e i . e ., A - B \neq B - A \end{array}$

Q10. If matrix $A B = O$ , then $A = O$ or $B = O$ or both $A$ and $B$ are null matrices.

Sol.

$\begin{array}{l} F a l s e . \\ Since f o r a n y t w o n o n - z e r o m a t r i c e s A a n d B, w e m a y g e t A B = 0 . \end{array}$

Q11. Transpose of a column matrix is a column matrix.

Sol.

Q12. If $A$ and $B$ are two square matrices of the same order, then $A B = B A$ .

Sol.

$\begin{array}{l} F a l s e . \\ F o r t w o s q u a r e m a t r i c e s A a n d B, A B = B A i s n o t a l w a y s t r u e . \end{array}$

Q13. If each of the three matrices of the same order is symmetric, then their sum is a symmetric matrix.

Sol.

$\begin{array}{l} T r u e . \\ L e t A, B, a n d C b e t h r e e m a t r i c e s o f t h e s a m e o r d e r . \\ G i v e n t h a t A^{'} = A, B^{'} = B, a n d C^{'} = C \\ L e t P = A + B + C \\ \Rightarrow P^{'} = {(A + B + C)}^{'} \\ = A^{'} + B^{'} + C^{'} = A + B + C = P \\ S o, A + B + C i s a l s o a s y m m e t r i c m a t r i x . \end{array}$

Q14. If $A$ and $B$ are any two matrices of the same order, then ${(A B)}^{'} = A^{'} B^{'}$ .

Sol.

$\begin{array}{l} F a l s e . \\ Since {(A B)}^{'} = B^{'} A^{'} . \end{array}$

Q15. If ${(A B)}^{'} = B^{'} A^{'}$ , where $A$ and $B$ are not square matrices, then the number of rows in $A$ is equal to the number of columns in $B$ and the number of columns in $A$ is equal to the number of rows in $B$ .

Sol.

$\begin{array}{l} T r u e . \\ L e t A = {[a_{i j}]}_{m \times n} a n d B = {[b_{i j}]}_{p \times q} \\ A B i s d e f i n e d w h e n n = P \\ ∴ O r d e r o f A B = m \times q \\ \Rightarrow O r d e r o f {(A B)}^{'} = q \times m \\ O r d e r o f B^{'} i s q \times p a n d o r d e r o f A^{'} i s n \times m \\ ∴ B^{'} A^{'} i s d e f i n e d w h e n P = n \\ a n d t h e o r d e r o f B^{'} A^{'} i s q \times m \\ H e n c e, O r d e r o f {(A B)}^{'} = O r d e r o f B^{'} A^{'} i . e ., q \times m . \end{array}$

Q16. If $A$ , $B$ , and $C$ are square matrices of the same order, then $A B = A C$ always implies that $B = C$ .

Sol.

$\begin{array}{l} F a l s e . \\ A = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], B = [\begin{matrix} 0 & 0 \\ 2 & 0 \end{matrix}] a n d C = [\begin{matrix} 0 & 0 \\ 3 & 4 \end{matrix}] \\ ∴ A B = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} 0 & 0 \\ 2 & 0 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] \\ ∴ A C = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} 0 & 0 \\ 3 & 4 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] \\ H e r e A B = A C = 0 b u t B \neq C . \end{array}$

Q17. $A A^{'}$ is always a symmetric matrix for any matrix $A$ .

Sol.

$\begin{array}{l} T r u e . \\ L e t P = A A^{'} \\ P^{'} = {(A A^{'})}^{'} \\ = {(A^{'})}^{'} . A^{'} [∵ {(A B)}^{'} = B^{'} A^{'}] \\ = A A^{'} = P \\ S o, P i s a s y m m e t r i c m a t r i x . \\ H e n c e, A A^{'} i s a l w a y s a s y m m e t r i c m a t r i x . \end{array}$

Q18. If

then $A B$ and $B A$ are defined and equal.

Sol.

Q19. If $A$ is a skew symmetric matrix, then $A^{2}$ is a symmetric matrix.

Sol.

$\begin{array}{l} T r u e . \\ {(A^{2})}^{'} = {(A^{'})}^{2} \\ = {[- A]}^{2} [∵ A^{'} = - A] \\ = A^{2} \\ S o, A^{2} i s a s y m m e t r i c m a t r i x . \end{array}$

Q20. ${(A B)}^{- 1} = A^{- 1} B^{- 1}$ , where $A$ and $B$ are invertible matrices satisfying the commutative property with respect to multiplication.

Sol.

$\begin{array}{l} T r u e . \\ I f A a n d B a r e i n v e r t i b l e m a t r i c e s o f t h e s a m e o r d e r . \\ ∴ {(A B)}^{- 1} = {(B A)}^{- 1} [∵ A B = B A] \\ B u t {(A B)}^{- 1} = A^{- 1} B^{- 1} [G i v e n] \\ ∴ {(B A)}^{- 1} = B^{- 1} A^{- 1} \\ A^{- 1} B^{- 1} = B^{- 1} A^{- 1} \\ ∴ A a n d B s a t i s f y c o m m u t a t i v e p r o p e r t y w . r . t m u l t i p l i c a t i o n . \end{array}$

Commonly asked questions

Q:

A matrix denotes a number.

Q:

Matrices of any order can be added.

Q:

Matrices of different orders cannot be subtracted

24th June 2022 (second shift)

Commonly asked questions

Q:

Let x * y = x² + y³ and (x * 1) * 1 = x * (1 * 1). Then a value of 2 $s i n^{- 1} (\frac{x^{4} + x^{2} - 2}{x^{4} + x^{2} + 2})$ is.

Q:

The sum of all the real roots of the equation $(e^{2 x} - 4) (6 e^{2 x} - 5 e^{x} + 1) = 0 i s$

Q:

Let the system of linear equations

x + y + $α z = 2$

3x + y + z = 4

x + 2z = 1

have a unique solution $(x^{*}, y^{*}, z^{*}) .$ If $(α, x^{*}), (y^{*}, α) a n d (x^{*}, - y^{*})$ are collinear points, then the sum of absolute values of all possible values of $α$ is

JEE Mains Solutions 2022,28th june , Maths,Second shift

Commonly asked questions

Q:

Let R₁ = ${(a, b) \in N \times N : | a - b | \leq 13} a n d$

R₂ = ${(a, b) \in N \times N : | a - b | \neq 13} .$ Then on N :

Q:

let f(x) be a quadratic polynomial such that f(-2) + f(3) = 0. If one of the roots of f(x) = 0 is -1, then the sum of the roots of f(x) = 0 is equal to:

Q:

The number of ways to distribute 30 identical candies among four children C₁, C₂, C₃ and C₄ so that C₂ receives atleast 4 and atmost 7 candies, C₃ receives atleast 2 and atmost 6 candies, is equal to:

Maths NCERT Exemplar Solutions Class 12th Chapter Three: Overview, Questions, Preparation

Matrices Long Answers Type Questions

Commonly asked questions

Matrices Short Answers Type Questions

Commonly asked questions

Matrices Objective Type Questions

Commonly asked questions

Matrices Fill in the blanks Type Questions

Commonly asked questions

Matrices True or False Type Questions

Commonly asked questions

24th June 2022 (second shift)

Commonly asked questions

JEE Mains Solutions 2022,28th june , Maths,Second shift

Commonly asked questions

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