Maths Matrices

Let A = [a? ]? Sum of diagonal elements of A.A? is Tr (A.A? ) = ∑? ∑? a? ² = 9.
where each a? ∈ {0, 1, 2, 3}.
Case I: One of a? = 3 and rest are 0. (3²=9). There are? C? = 9 ways.
Case II: Two of a? are 2, one is 1, and rest are 0. (2² + 2² + 1² = 9). There are? C? *? C? = 36 * 7 = 252 ways.
Case III: One of a? = 2, five are 1, and rest are 0. (2² + 1²+1²+1²+1²+1² = 9). There are? C? *? C? = 9 * 56 = 504 ways.
Case IV: All nine a? = 1. (1² * 9 = 9). There is 1 way.
Total = 9 + 252 + 504 + 1 = 766.

A = [i, -i], [-i, i]
A² = [-2, 2], [2, -2]
A? = [8, -8], [-8, 8]
A? = [-128, 128], [128, -128]
A? [x, y]? =?
-128x + 128y = 8 ⇒ -16x + 16y = 1 ⇒ x - y = -1/16 (I)
128x - 128y = 64 ⇒ 16x - 16y = 8 ⇒ x - y = 1/2 (II)
System is inconsistent hence No solution

A = 1/3 [ 1; 1 ω ω² 1 ω² ω ]
A² = A * A = 1/9 [ . ]
(The calculation in the image shows A² is the identity matrix, let's verify)
A² leads to I (Identity matrix).
So A² = I.
A³ = A² * A = I * A = A.
A? = (A²)² = I² = I.
A³? = (A²)¹? = I¹? = I.

[a? ] = [ 1 ] [-1 b? ]
[a? ] [√3 k] [ k b? ] (This is likely incorrect OCR, should be a 2x1 result)
The solution seems to derive from a matrix multiplication:
[√3a? ] = [ 1 ] [b? ]
[√3a? ] [√3 k] [b? ]
This leads to:
b? - b? = √3a?
b? + kb? = √3a?
Also given: a? ² + a? ² = (2/3) (b? ² + b? ²).
Squaring and adding the two derived equations and comparing with the given condition leads to k=1.

Given matrices A = [[a, b], [c, d]] and B = [[α], [β]] where B ≠ [[0], [0]].
The product AB is:
AB = [[a, b], [c, d]] * [[α], [β]] = [[aα + bβ], [cα + dβ]]

From the problem statement AB = B, we have:
aα + bβ = α (i)
cα + dβ = β (ii)

Rearranging these equations:
(a - 1)α + bβ = 0
cα + (d - 1)β = 0

For this system of linear equations to have a non-trivial solution (since B is not the zero matrix), the determinant of the coefficient matrix must be zero.
det([[a-1, b], [c, d-1]]) = 0

(a - 1)(d - 1) - bc = 0
ad - a - d + 1 - bc = 0
ad - bc = a + d - 1
The provided text jumps to the conclusion ad - bc = 2020.

The truth table for the logical expression (p ∧ q) → (p → q) is as follows:

The final column shows that the expression is a tautology, meaning it is always true regardless of the truth values of p and q.

1 = (2-1)¹ (The n is likely 1).
3? = (7-4)³ (This seems to be a pattern matching (a-b)^c).
4²? = (12-8)? ! = 4²?
The blank space must be (5-3)² = 2² = 4.

Evaluate the integral:
∫ (2x-1)cos (√ (4x²-4x+6) / √ (4x²-4x+6) dx
∫ (2x-1)cos (√ (2x-1)²+5) / √ (2x-1)²+5) dx

Let (2x-1)² + 5 = t².
Differentiating both sides:
2 (2x-1)*2 dx = 2t dt
2 (2x-1) dx = t dt
(2x-1) dx = (t/2) dt

Substitute into the integral:
∫ cos (t)/t * (t/2) dt
= 1/2 ∫ cos (t) dt
= 1/2 sin (t) + C
= 1/2 sin (√ (2x-1)²+5) + C
= 1/2 sin (√ (4x²-4x+6) + C

Given the equations:
t? (A + 2B) = -1 which expands to t? (A) + 2t? (B) = -1 . (I)
t? (2A - B) = 3 which expands to 2t? (A) - t? (B) = 3 . (II)

Solving equations (I) and (II) simultaneously, we get:
t? (A) = 1
t? (B) = -1

Therefore, t? (A) - t? (B) = 1 - (-1) = 2.