Table of Contents

Toggle**Matrix**

A matrix is a rectangular arrangement of numbers (real or Complex) which may be represented as

\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}

Matrix is enclosed by [] or ()

The compact form of the above matrix is represented by [a_{ij}] _{m×n} or A = [a_{ij}]

## Order of a Matrix

The order of the matrix is the number of rows and columns present in the matrix.

If the matrix has m rows and n columns, then the order is m×n

## Type of matrix

**Row Matrix**: A matrix with only one row is called a row matrix. It has 1 row and multiple columns.**Column Matrix**: A matrix with only one column is called a column matrix. It has 1 column and multiple rows.**Square Matrix**: A matrix in which the number of rows is equal to the number of columns is called a square matrix.**Diagonal Matrix**: A square matrix where all the elements outside the main diagonal are zero is called a diagonal matrix.**Identity Matrix (I)**: A special diagonal matrix where all the diagonal elements are 1 and all other elements are 0 is called an identity matrix.**Zero Matrix (O)**: A matrix where all the elements are zero is called a zero matrix or a null matrix.**Scalar Matrix**: A diagonal matrix where all the diagonal elements are the same is called a scalar matrix.**Symmetric Matrix**: A square matrix that is equal to its transpose is called a symmetric matrix.**Skew-Symmetric Matrix**: A square matrix whose transpose is equal to the negation of itself is called a skew-symmetric matrix.

## Algebra of Matrix

- Addition of Matrices
- Subtraction of Matrices
- Multiplication of a Matrix by a Scalar
- Multiplication of Matrix

Let A = \begin{bmatrix}a & b \\c & d \end{bmatrix} and B = \begin{bmatrix}p & q \\r & s \end{bmatrix}

### Addition of Matrix

A + B = \begin{bmatrix}a & b \\c & d \end{bmatrix} + \begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}a + p & b + q \\c + r & d + s \end{bmatrix}

**Subtraction of Matrix**

A – B = \begin{bmatrix}a & b \\c & d \end{bmatrix} – \begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}a - p & b - q \\c - r & d - s \end{bmatrix}

### Multiplication of Matrix by a Scalar

kA = k \begin{bmatrix}a & b \\c & d \end{bmatrix} = \begin{bmatrix}ka & kb \\kc & kd \end{bmatrix}

Where k is the scalar value or constant number

### Multiplication of Matrices

A×B=\begin{bmatrix}a & b \\c & d \end{bmatrix}×\begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}ap + br & aq+bs \\cp+dr & cq+ds \end{bmatrix}

B×A=\begin{bmatrix}p & q \\r & s \end{bmatrix}×\begin{bmatrix}a & b \\c & d \end{bmatrix} = \begin{bmatrix}pa+qc & pb+qd \\ra+sc & rb+sd \end{bmatrix}

Note: A×B ≠ B×A

## Transpose of a Matrix

Let A be a matrix of order m × n. Then, the n × m matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by A’ or A^{T}.

### Properties of transpose

For any two matrices A and B of suitable orders.

- (A’)’ = A
- (A ± B)’ = A’ ± B’
- (kA)’ = kA’
- (AB)’ = B’A’
- (ABC)’ = C’B’A

**Symmetric and Skew-Symmetric Matrices**

- A square matrix A is said to be symmetric if A’ = A.
- A square matrix A is said to be skew-symmetric if A’ = -A

### Properties

If A is a square matrix, then

- A + A’ is a symmetric matrix
- A – A’ is a skew-symmetric matrix
- The sum of symmetric and skew-symmetric matrix = ½ (A + A’) + ½ (A – A’)

If A and B are symmetric matrices of the same order, then

- AB – BA is a skew-symmetric matrix and
- AB + BA is symmetric matrix

## Some special types of Matrix

**Orthogonal Matrix:** A square matrix of order n is said to be orthogonal if AA’ = I_{n} = A’A.

**Properties of Orthogonal Matrix**

- If A is an orthogonal matrix, then A’ is also an orthogonal matrix.
- For any two orthogonal matrices A and B. AB and BA is also orthogonal matrix.
- If A is an orthogonal matrix, then A
^{-1}is also orthogonal matrix.