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Using Matrices

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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.

Given below is an example of matrix with three rows and three columns. They are commonly written in brackets.
 A = $\bigl(\begin{smallmatrix}
1 & 2 & -9\\
 7& -8 &4 \\
0 & 9 &8
\end{smallmatrix}\bigr)$
The above matrix is a 3 * 3 matrix because it has three rows and three columns. When describing matrices the format to be used is rows * columns.

Matrix Definition

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Matrix is a array of numbers inside a rectangular or square grid. Array can be of numeric or algebraic quantities subjected to mathematical operations. It is a two dimensional array which has rows and columns to represent elements. In matrix we denote row by 'i' and column by 'j' and a matrix is usually represented by a capital letter. Matrices are based on systems of linear equations.

Matrix Operations

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Operations that can be performed on matrix are explained below.

1. Equality of matrices: Two matrices of the same size are defined to be equal if their corresponding elements are equal.
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. Addition of matrices: 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.
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. Negative of a matrix: 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.
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

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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

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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
 

Symmetric Matrix

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A square matrix A is called symmetric if it is equal to its transpose. That is A = 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

Invertible Matrix

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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 $\begin{bmatrix}
 -2& 1\\
-3 & 2
\end{bmatrix}$

= $\begin{bmatrix}
 2& -1\\
3 & -2
\end{bmatrix}$
 

Skew Symmetric Matrix

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A square matrix A is called a skew symmetric matrix if its negative is equal to its transpose.
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