What is Matrix in Math? (Introduction, Types & Matrices Operations)

Matrix is an important topic in mathematics. In this post, we are going to discuss these points.

In 1858, a British Mathematician Arthur Cayley was first developed “Theory of Matrices”. Arthur Cayley was also the organizer of the Modern British School of Pure Mathematics.

In childhood, he loved to solve complex mathematical problems for pleasure and shined himself in French, German, Italian, Greek and Mathematics from Trinity College Cambridge.

What is the Matrix?

Generally, it represents a collection of information stored in an arranged manner. Mathematically, it states to a set of numbers, variables or functions arranged in rows and columns. Matrices are represented by the capital English alphabet like A, B, C……, etc.

For example,

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 3 & 4 & 5 \\ 6 & 7 & 8 \\ 9 & 2 & 3 \end{array}} \right]$

In above example, Matrix A has 3 rows and 3 columns.

Application of Matrices

Matrices are used in various branches of science, some of its applications are:

1. To solve the system of linear equations
2. Computer Graphics
3. Physics
4. Cryptography
5. Graph Theory

Order of a Matrix

It is defined by the number of rows and columns in a matrix.

Order of a matrix = number of rows × number of columns

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 2 & 5 & 8 \\ 8 & 2 & 3 \end{array}} \right]$

In above example, number of rows is 3 while number of columns is also 3, therefore,

Order of matrix A is 3 × 3.

Types of Matrices

1. Rows Matrix

If a matrix has only one row then it is called a row matrix. For example,

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \end{array}} \right]$

It is a row matrix of order 1 by 3.

2. Columns Matrix

If a matrix has only one column then it is called a column matrix. For example,

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \end{array}} \right]$

It is column matrix of order 3 by 1.

3. Null or Zero Matrix

If all the entries of a matrix are zero then it is called a Null or zero matrix. For example,

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 0 & 0 \\ 0 & 0 \end{array}} \right]$

It is null matrix of order 2 by 2. A null or zero matrix is denoted by ‘O’.

4. Square Matrix

In a matrix, if the number of rows is equal to the number of columns, then it is called a Square Matrix. For example, if a matrix has 2 rows and 2 columns then it is called a Square Matrix as given below

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 3 \\ 2 & 4 \end{array}} \right]$

5. Rectangular Matrix

In a matrix, if the number of rows is not equal to the number of columns, then it is called a Rectangular Matrix. For example, if a matrix has 2 rows and 3 columns then it is called a Rectangular Matrix as given below.

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 5 & 6 & 7 \end{array}} \right]$

6. Diagonal Matrix

If all the elements of a square matrix are zero except those in the main diagonal, then it is called a Diagonal Matrix. However, few elements of the main diagonal can be zero but not all elements. For example,

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 3 \end{array}} \right]$

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{array}} \right]$

7. Scalar Matrix

If all the diagonal elements of a diagonal matrix are same, then it is called a Scalar Matrix. For example,

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}} \right]$

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 4 & 0 \\ 0 & 4 \end{array}} \right]$

8. Unit or Identity Matrix

If each diagonal element of a diagonal matrix is 1, then it is called a Unit or Identity Matrix. For example,

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}} \right]$

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & 1 \end{array}} \right]$

9. Negative of a Matrix

A Negative matrix is obtained by replacing the signs of its all entries. Consider a matrix A and let’s change it into negative matrix –A as,

if

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 3 & {-2} & 5 \\ {-3} & 5 & 6 \end{array}} \right]$

then

$\displaystyle -A=\left[ {\begin{array}{*{20}{c}} {-1} & {-2} & {-3} \\ {-3} & 2 & {-5} \\ 3 & {-5} & {-6} \end{array}} \right]$

10. Transpose of a Matrix

A transpose of a matrix is obtained by interchanging all of its rows into columns or columns into rows. It is denoted by $\displaystyle {{A}^{t}}$ or $\displaystyle {{A}^{'}}$. For example,

If

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 4 & 3 & 5 \\ 2 & 6 & 2 \end{array}} \right]\,$

then

$\displaystyle {{A}^{'}} =\left[ {\begin{array}{*{20}{c}} 1 & 4 & 2 \\ 2 & 3 & 6 \\ 3 & 5 & 2 \end{array}} \right]$

11. Symmetric Matrix

A square matrix is said to be Symmetric if it is equal to its transpose. For example,

If

$\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{array}} \right]\,$

then

$\displaystyle {{A}^{t}}=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{array}} \right]\,\,\,=A\,\,$

Therefore, A is Symmetric.

If $\displaystyle B=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 2 & 1 & 5 \\ 3 & 8 & 4 \end{array}} \right]$ then $\displaystyle {{B}^{t}}=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 2 & 1 & 8 \\ 3 & 5 & 4 \end{array}} \right]\,\,\,\ne B\,\,$

Hence, B is not symmetric.

12. Skew-Symmetric Matrix

A square matrix is said to be Skew-Symmetric if its transpose is equal to the negative of this matrix i.e. $\displaystyle {{A}^{t}}=-A$. For example,

If  $\displaystyle A=\left[ {\begin{array}{*{20}{c}} 0 & {-5} & 2 \\ 5 & 0 & {-4} \\ {-2} & 4 & 0 \end{array}} \right]\,\,$

then

$\displaystyle {{A}^{t}}=\left[ {\begin{array}{*{20}{c}} 0 & 5 & {-2} \\ {-5} & 0 & 4 \\ 2 & {-4} & 0 \end{array}} \right]\,\,=(-1)\,\,\,\left[ {\begin{array}{*{20}{c}} 0 & {-5} & 2 \\ 5 & 0 & {-4} \\ {-2} & 4 & 0 \end{array}} \right]$

$\displaystyle {{A}^{t}}=-A$

Hence, Matrix A is Skew-Symmetric.

Operations on Matrices

The addition to two matrices A and B will be possible if they have the same orders. Addition of two matrices A and B is denoted by A + B. For example,

If $\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 5 & 6 & 7 \end{array}} \right]$ and $\displaystyle B=\left[ {\begin{array}{*{20}{c}} 2 & 5 & 3 \\ 4 & 1 & 8 \end{array}} \right]$

Then,

$\displaystyle A+B=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 5 & 6 & 7 \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} 2 & 5 & 3 \\ 4 & 1 & 8 \end{array}} \right]$

$\displaystyle =\left[ {\begin{array}{*{20}{c}} {1+2} & {2+5} & {3+3} \\ {5+4} & {6+1} & {7+8} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 3 & 7 & 6 \\ 9 & 7 & {15} \end{array}} \right]$

Subtraction of Matrices:

The subtraction of one matrix from another matrix will be possible if they have the same orders. Subtraction of two matrices A and B is denoted by A – B. For example,

If $\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 5 & 6 & 7 \end{array}} \right]$ and $\displaystyle B=\left[ {\begin{array}{*{20}{c}} 2 & 5 & 3 \\ 4 & 1 & 8 \end{array}} \right]$

Then,

$\displaystyle A-B=\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 \\ 5 & 6 & 7 \end{array}} \right]-\left[ {\begin{array}{*{20}{c}} 2 & 5 & 3 \\ 4 & 1 & 8 \end{array}} \right]$

$\displaystyle =\left[ {\begin{array}{*{20}{c}} {1-2} & {2-5} & {3-3} \\ {5-4} & {6-1} & {7-8} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {-1} & {-3} & 0 \\ 1 & 5 & {-1} \end{array}} \right]$

Product of Matrices:

The product of two matrices A and B will be possible if the number of columns of a Matrix A is equal to the number of rows of another Matrix B. For example,

If $\displaystyle A=\left[ {\begin{array}{*{20}{c}} 1 & 2 \\ 3 & 4 \end{array}} \right]$ and $\displaystyle B=\left[ {\begin{array}{*{20}{c}} 3 & 2 \\ 1 & 4 \end{array}} \right]$

Then,

$\displaystyle AB=\left[ {\begin{array}{*{20}{c}} 1 & 2 \\ 3 & 4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 3 & 2 \\ 1 & 4 \end{array}} \right]$

$\displaystyle AB=\left[ {\begin{array}{*{20}{c}} {1\times 3+2\times 1} & {1\times 2+2\times 4} \\ {3\times 3+4\times 1} & {3\times 2+4\times 4} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {3+2} & {2+8} \\ {9+4} & {6+16} \end{array}} \right]$

$\displaystyle AB=\left[ {\begin{array}{*{20}{c}} 5 & {10} \\ {13} & {22} \end{array}} \right]$

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