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Matrices Class 12 Mathematics

## 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

• 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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