Question

What is matrix theory?

Answer 1

Hint: In this question, we are going to see the following points in detail
Definition of Matrix
Types of Matrices
Matrix operations
Matrix is a rectangular array of M X N elements used to express the collection of data. Matrix contains m horizontal lines and n vertical lines. Here M X N is the order of the matrix. A matrix is denoted by or . The types of matrix are Identity matrix, row matrix, column matrix, symmetric, skew-symmetric etc. We have discussed the matrix theory in detail below.

Complete step-by-step answer:
In this question, we are going to discuss the matrix theory.
First of all, let us see the definition of matrix and some examples of matrices.
$ \Rightarrow $Matrix Definition:
Matrix is a rectangular array of $m \times n$ elements used to express the collection of data. Matrix contains m horizontal lines and n vertical lines. Here, $m \times n$ is the order of the matrix. A matrix is denoted by $\left( {} \right)$ or $\left[ {} \right]$.
A $m \times n$ matrix is denoted by:
\[\left[ {\begin{array}{*{20}{c}}
  {{a_{11}}}&{{a_{12}}}&{...}&{{a_{1n}}} \\
  {{a_{21}}}&{{a_{22}}}&{...}&{{a_{2n}}} \\
   \vdots & \vdots & \vdots & \vdots \\
  {{a_{m1}}}&{{a_{m2}}}&{...}&{{a_{mn}}}
\end{array}} \right]\]
The numbers ${a_{11}},{a_{12}},{a_{\,21}},...$ are known as the elements of the matrix. It is denoted by
$A = \left[ {{a_{ij}}} \right]$
$ \Rightarrow $Types of Matrices:
Symmetric Matrix
A square matrix $A = \left[ {{a_{ij}}} \right]$ is said to be symmetric matrix if ${a_{ij}} = {a_{ji}}$, for all the values of i and j.
Example: $A = \left[ {\begin{array}{*{20}{c}}
  1&2&3 \\
  2&4&5 \\
  3&5&2
\end{array}} \right]$
Skew – Symmetric Matrix
A square matrix $A = \left[ {{a_{ij}}} \right]$ is said to be skew- symmetric matrix if its transpose is negative of itself, that is ${A^T} = - A$.
Example: $A = \left[ {\begin{array}{*{20}{c}}
  0&2&{ - 7} \\
  { - 2}&0&3 \\
  7&{ - 3}&0
\end{array}} \right]$
Identity Matrix
If all the diagonal elements of a square matrix are equal to 1 and all other elements are 0, then the matrix is said to be the Identity matrix. It is denoted by $I$.
Example: $I = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right]$

Note: Matrix operations: Matrix operations has three main algebraic operations
Addition of matrices
Subtraction of matrices
Multiplication of matrices
$ \Rightarrow $If $A{\left[ {{a_{ij}}} \right]_{m \times n}}$ and $B{\left[ {{b_{ij}}} \right]_{m \times n}}$ are two given matrices then their sum $A + B$ is given by
$A + B = {\left[ {{a_{ij}} + {b_{ij}}} \right]_{m \times n}}$.
$ \Rightarrow $If A and B are two matrices of same order then their subtraction is given by
$A - B = A + \left( { - B} \right)$.
$ \Rightarrow $If A and B are matrices, then their product will only be defined if the number of columns in A will be equal to the number of rows in B.
If $A{\left[ {{a_{ij}}} \right]_{m \times n}}$ and $B{\left[ {{b_{ij}}} \right]_{n \times p}}$ are two matrices then their product is given by $C{\left[ {{c_{ij}}} \right]_{m \times p}}$.