Matrices Class 12 NCERT Solutions – Easy Step-by-Step Answers

Here is a quick summary of the main concepts covered in the Matrices Class 12:

• Class 12th Maths NCERT Solutions cover the definition of a matrix. It is an ordered rectangular array of functions or numbers. If a matrix has m rows and n columns, the matrix order is - $m\times n$
• It covers column matrix, row matrix, square matrix, diagonal matrix, scalar, and identity matrix.
• Also, the zero matrix, symmetric matrix, skew-symmetric matrix, and inverse of a square matrix.

Check here for NCERT solutions of all Class 12 Maths chapters with free PDF, important topics. You will also get the weightage information for the chapters.

Those who are looking to download the free Matrices Class 12 NCERT PDF can find the link below. The students must download it for their exam preparation. The step-by-step solutions will help them understand the concepts clearly and score high in the exams.

Class 12 Math Chapter 3 Matrices Solution: Free PDF Download

While preparing the Class 12th Maths NCERT Solutions Matrices, students should focus on the core concepts of the matrices, including subtraction, addition, multiplication, and finding the determinant and inverse of matrices. See below the topics covered in Maths Class 12 NCERT Solutions Matrices:

Maths Class 12 Matrices Weightage in JEE Mains

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The class 12 Matrices Chapter focuses on several important topics, such as the Type of Matrices, zero, symmetric, and asymmetric Matrices, the Transpose and Inverse of Matrices, and other Matrices operations. Students will encounter different problems based on these concepts in the exercises. Each exercise deals with different concepts, Class 12 Matrices Exercise 3.1 deals with the basics of Matrix, its types and fundamentals. Class 12 Matrices Exercise 3.2  deals with operations of Matrix, its multiplication both scaler and vector and more advanced concepts. Class 12 Matrices Exercise 3.3 deals with problems finding the transpose and inverse of Matrix. Students can check Exercise-wise Class 12 Chapter 3 Matrices NCERT Solutions below;

Matrices exercise 3.1 solutions focuses on the basic concepts of matrices. Definition, types of Matrices, and basic operations of matrices are several topics discussed in the solutions of Matrices Class 12 Exercise 3.1. Chapter 3 Matrices Exercise 3.1 includes 10 Questions (7 Short Answers, 3 MCQs). Students can find the solution of exercise 3.1 below;

Matrices Exercise 3.1 Solutions

Q4.Construct a 2 × 2 matrix,  $A=\left[a_{ij}\right]$ , whose elements are given by: $a_{ij}=\frac{{\left(i+j\right)}^2}{2}$ (ii)  $a_{ij}=\frac{i}{j}$ (iii)  $a_{ij}=\frac{{\left(i+2j\right)}^2}{2}$

A.4. (E) (i) aij ${\frac{\left(i+j\right)}{2}}^2$ such that i = 1, 2 and j = 1 × 2 for 2 × 2 matrix Therefore a11 = $\frac{{\left(1+1\right)}^2}{2}=\frac{2^2}{2}=2$ A 2×2 = $\left[\begin{array}{cc}\hfill a_{11}\hfill & \hfill a_{12}\hfill \\ \hfill a_{21}\hfill & \hfill a_{22}\hfill \end{array}\right]$ a12 = $\frac{{\left(1+2\right)}^2}{2}=\frac{3^2}{2}=\frac{9}{2}$ a21 $\frac{{\left(2+1\right)}^2}{2}=\frac{3^2}{2}=\frac{9}{2}$  $\left[\begin{array}{cc}\hfill 2\hfill & \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\hfill \\ \hfill \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\hfill & \hfill 8\hfill \end{array}\right]$ a22 = ${\frac{\left(2+2\right)}{2}}^2=\frac{4^2}{2}=\frac{16}{2}=8$

Matrices Important Formulae for CBSE and Competitive Exams

Students can check important formulae and basic concepts of the Matrices chapter below for a better understanding of questions. Students can also use these formulae to solve exercises.

Matrix Basics

• Order of a Matrix:
  • If a matrix has $m$ rows and $n$ columns, its order is $m \times n$.
• Elements:
  • $A = [a_{ij}]$, where $a_{ij}$ represents the element in the $i^{\text{th}}$ row and $j^{\text{th}}$ column.

Matrix Operations

• Addition/Subtraction:

  $$(A \pm B)_{ij} = a_{ij} \pm b_{ij}$$

• Scalar Multiplication:

  $$(kA)_{ij} = k \cdot a_{ij}$$

• Matrix Multiplication:

  $$(AB)_{ij} = \sum_{k=1}^n a_{ik} \cdot b_{kj}$$
  • Condition: Number of columns in $A$ = Number of rows in $B$.

• Transpose of a Matrix:

  $$(A^T)_{ij} = a_{ji}$$

Properties of Matrix Operations

• Transpose Properties:

  $$(A^T)^T = A$$ $$(A + B)^T = A^T + B^T$$ $$(kA)^T = kA^T$$ $$(AB)^T = B^T A^T$$

• Symmetry:

  $$A^T = A \quad (\text{Symmetric})$$ $$A^T = -A \quad (\text{Skew-Symmetric})$$

• Identity Matrix Properties:

  $$AI = IA = A$$

Determinant and Inverse of Square Matrices

• Adjoint Formula for Inverse:

  $$A^{-1} = \frac{\text{Adj}(A)}{\det(A)}$$

• Determinant Property:

  $$\det(A^T) = \det(A)$$

Applications of Matrcies

• Solving Linear Equations: $$AX = B \implies X = A^{-1}B$$
• Consistency of Linear Equations:
  • If $\det(A) \neq 0$, the system has a unique solution.