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Download the PDF## Matrices Class 12 Mathematics

### 1. Basic Concepts

#### 1.1 Definition

A matrix is a collection of numbers arranged into a fixed number of rows and columns. It looks something like this:

$$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$

Here, \(a_{11}, a_{12}, a_{21}, a_{22}\) are the elements of the matrix.

#### 1.2 Types of Matrices

**Square Matrix:**A matrix with the same number of rows and columns.**Rectangular Matrix:**A matrix with a different number of rows and columns.**Diagonal Matrix:**A square matrix where all elements outside the main diagonal are zero.**Scalar Matrix:**A diagonal matrix where all the main diagonal elements are the same.**Identity Matrix:**A diagonal matrix where all the main diagonal elements are 1.**Zero Matrix:**A matrix where all elements are zero.

### 2. Important Properties

#### 2.1 Addition and Subtraction

**Closure Property:**The sum or difference of two matrices of the same dimensions is a matrix of the same dimensions.**Commutative Property of Addition:**\(A + B = B + A\).**Associative Property of Addition:**\(A + (B + C) = (A + B) + C\).**Existence of Additive Identity:**Adding a zero matrix to any matrix \(A\) leaves \(A\) unchanged.

#### 2.2 Multiplication

**Closure Property:**The product of two matrices is defined if the number of columns in the first matrix is the same as the number of rows in the second.**Associative Property:**\(A(BC) = (AB)C\).**Distributive Property:**\(A(B + C) = AB + AC\) and \((A + B)C = AC + BC\).**Existence of Multiplicative Identity:**For any square matrix \(A\), \(AI = IA = A\), where \(I\) is the identity matrix of the same order as \(A\).

#### 2.3 Transpose of a Matrix

**(A^T)^T = A**.**(A + B)^T = A^T + B^T**.**(AB)^T = B^T A^T**.

#### 2.4 Inverse of a Matrix

**(A^{-1})^{-1} = A**.**(AB)^{-1} = B^{-1}A^{-1}**.

#### 2.5 Determinant

**det(A^T) = det(A)**.**det(AB) = det(A)det(B)**.**If A is invertible, then det(A^{-1}) = 1/det(A)**.

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