Matrices are a set of numbers or we can say it is a collection of numbers arranged in rows and columns. The numbers which we are using in matrices are real numbers, complex numbers can also be used. With the help of matrices we can solve many problems like, when a lot of repetitive mathematical actions are to be done and when the operator (Can be anything, for example, Computer) is programmed to repeat the same kind of operation with sets of numbers that are taken from the matrix or matrices. Matrices makes life easier as all the number crunching can be done with the help of a computer program. They also help in to encrypt and decrypt codes. |

Operations that can be performed on matrix are explained below.

1.

*: Two matrices of the same size are defined to be equal if their corresponding elements are equal.*

**Equality of matrices**Thus A = $\begin{bmatrix}

a &b &c \\

x& y &z

\end{bmatrix}$

and B = $\begin{bmatrix}

0 &1 &3 \\

2& 4 &7

\end{bmatrix}$

Then A and B are equal if and only if a = 0, b = 1, c = 3, x = 2, y = 4, and z = 7

When A and B are equal we write A = B

2.

**: If A and B are matrices of the same size, then their sum denoted by A + B is defined to be the matrix of the same size A and B got by adding the corresponding elements of A and B.**

__Addition of matrices__Thus if A = $\begin{bmatrix}

a &b &c \\

x& y &z

\end{bmatrix}$ and

B = $\begin{bmatrix}

0 &1 &3 \\

2& 4 &7

\end{bmatrix}$

then A + B = $\begin{bmatrix}

a + 0 & b + 1 &c + 3 \\

x +2&y + 4 & z +7

\end{bmatrix}$

*3.***: If A is any matrix, then the matrix whose elements are the corresponding elements of A taken with a negative sign is called the negative of A and is denoted as -A.**

__Negative of a matrix__Thus if A = $\begin{bmatrix}

1 &2 &7 \\

0& -4 &13

\end{bmatrix}$, - A =$\begin{bmatrix}

-1 &-2 &-7 \\

0& 4 &-13

\end{bmatrix}$

It is to be noted that A + ( - A ) is the zero matrix of the same size as A.

*4.*

__Matrix multiplication____:__Two matrices A and B can be multiplied to form the product AB if and only if the number of rows of B is equal to the number of columns of A. If A is a m * p order matrix and B is a p * n order matrix, then only the product AB, a matrix of order m * n is defined.

Diagonal matrix is a square matrix in which all the entries off the main diagonal are 0. Mathematically, A = [A$_{ij}$] is a diagonal matrix if and only if a$_{ij}$ = 0 for every i and j such that i $\neq$ j. A square matrix in which the entries outside the main diagonal are all zero is called a diagonal matrix. Sum of the entries on the main diagonal of a square matrix is known as the trace of that matrix.

Identity matrix is a scalar matrix in which all of the diagonal elements are unity.

Identity matrix is the identity for matrix multiplication.

Here IA = A and AI = A for all matrices A.

Given below are the four identity matrices.

I$_{1}$ = [1],

I$_{2}$ = $\begin{bmatrix}

1 &0 \\

0& 1

\end{bmatrix}$

I$_{3}$ = $\begin{bmatrix}

1 &0 & 0\\

0 & 1 & 0\\

0& 0 & 1

\end{bmatrix}$

I$_{4}$ = $\begin{bmatrix}

1& 0 &0 &0 \\

0& 1 & 0 & 0\\

0& 0 & 1 &0 \\

0& 0 & 0 & 1

\end{bmatrix}$

### Solved Example

**Question:**If A = $\begin{bmatrix}

9 & -8 & 7\\

-6& 5&-4 \\

3& -2 & 1

\end{bmatrix}$ and B = $\begin{bmatrix}

1 & 0& 0\\

0 & 1 & 0\\

0& 0 & 1

\end{bmatrix}$

Find AB and BA.

**Solution:**

AB = $\begin{bmatrix}

9 & -8 & 7\\

-6& 5&-4 \\

3& -2 & 1

\end{bmatrix}$ * $\begin{bmatrix}

1 & 0& 0\\

0 & 1 & 0\\

0& 0 & 1

\end{bmatrix}$

= $\begin{bmatrix}

9 + 0 + 0 &0 - 8 + 0 & 0 + 0 + 7\\

-6 + 0 + 0& 0 + 5 + 0 &0 + 0 -4 \\

3+ 0+ 0& 0 - 2 + 0 & 0 + 0 + 1

\end{bmatrix}$

= $\begin{bmatrix}

9 & -8 & 7\\

-6& 5&-4 \\

3& -2 & 1

\end{bmatrix}$

BA = $\begin{bmatrix}

1& 0 & 0\\

0 & 1 &0 \\

0& 0 & 1

\end{bmatrix}$ $\begin{bmatrix}

9& -8&7 \\

-6& 5& -4\\

3& -2&1

\end{bmatrix}$

= $\begin{bmatrix}

9 & -8 & 7\\

-6& 5&-4 \\

3& -2 & 1

\end{bmatrix}$

It should be noted that B is I, unit matrix of order 3.

Therefore, AI = IA = A

It is easy to see that a$_{ij}$ = a$_{ji}$ if a$_{ij}$ and a$_{ji}$ are respectively the elements in the ith row, jth column and jth row and ith column.

**Example:**A =$\begin{bmatrix}

1 &2& 4\\

2& 3&-4 \\

4&-4 & 0

\end{bmatrix}$

is a symmetric since A ' = $ \begin{bmatrix}

1 &2& 4\\

2& 3&-4 \\

4&-4 & 0

\end{bmatrix}$ = A

A n * n matrix A is called non singular if and only if there exists a n * n matrix B such that

AB = BA = I$_{n}$

where I$_{n}$ is the identity matrix. The matrix B is called the inverse matrix of A.

### Solved Example

**Question:**Let A = $\begin{bmatrix}

2& -1\\

3 & -2

\end{bmatrix}$

Find $A^{-1}$ if it exists

**Solution:**

| A | = (-4) + 3 = -1 $\neq$ 0

A is non singular. Hence A$^{-1}$ exists.

Adj A = $\begin{bmatrix}

-2& 1\\

-3 & 2

\end{bmatrix}$

$\therefore$ $A^{-1}$ = $\frac{1}{|A|}$ adj.A

= - 1 $\begin{bmatrix}

-2& 1\\

-3 & 2

\end{bmatrix}$

= $\begin{bmatrix}

2& -1\\

3 & -2

\end{bmatrix}$

That is A' = - A, that is a$_{ij}$ = -a$_{ji}$

Thus, in a skew-symmetric matrix

a$_{11}$ = -a$_{11}$ That is a$_{11}$ = 0

a$_{22}$ = -a$_{22}$ That is a$_{22}$ = 0

a$_{33}$ = -a$_{33}$ That is a$_{33}$ = 0 and so on

Thus, in a skew - symmetric matrix the elements on the principal diagonal are all equal to zero.

Thus the matrix

A = $\begin{bmatrix}

0 &2& 4\\

- 2& 0& 7\\

-4&-7 & 0

\end{bmatrix}$ is a skew - symmetric since

A' = $\begin{bmatrix}

0 &- 2& - 4\\

2& 0& -7\\

4&7 & 0

\end{bmatrix}$ = - A