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Equal Matrices

Learn about equal matrices concept. The matrix is the representation of elements in the form of rows and columns. If a$_{ij}$ is an element in matrix, then 'a' would represent the element of i$^{th}$ row and j$^{th}$ column. It is a rectangular representation of numbers of expressions. The general form of matrix is shown in the following diagram:

The number of rows and number of columns collectively denote the dimension of a matrix. If a matrix has m rows and n columns, then such matrix is said to have dimension or order of m x n. Have a look at the following example :

There are several concepts related to matrices. In this page, we are going to learn about equal matrices. We shall go ahead and understand the definition of equal matrices and examples based on those.

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Definition

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If a$_{ij}$ is an element of matrix A and b$_{ij}$ is element of matrix B, then the matrix A and B are equal only if a = b for all i and j values. Two matrices to be equal, their order must be the same. If two matrices have their corresponding elements equal, then they are called equal matrices.

If $\begin{bmatrix}a & b\\ c & d\end{bmatrix}$ = $\begin{bmatrix}p & q\\ r & s\end{bmatrix}$ then a = p, b = q, c = r and d = s.

Example:

$\begin{bmatrix}2 & 5 & 8\\ 1 & 2 & 0\end{bmatrix}$ = $\begin{bmatrix}2 & 5 & 8\\ 1 & 2 & 0\end{bmatrix}$

Note:

(i) Two equal matrices are exactly the same.

(ii) If two matrices are of the same order (no condition on elements) they are said to be comparable. Two matrices are said to be equal if they have the same order and their corresponding elements are equal.

Example
1) If $\begin{bmatrix}a & b & c\\ d & e & f\end{bmatrix}$ = $\begin{bmatrix}1 & 2 & 3\\ 4 & 5 & 6\end{bmatrix}$

then a = 1, b = 2, c = 3, d = 4, e = 5 and f = 6.

Equal Matrices Examples

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Below are some examples on equal matrices

Example 1 : If $\begin{bmatrix}a & 6\\ 10 & d\\ e & 0\end{bmatrix}$ = $\begin{bmatrix}2 & 6\\ 10 & -1\\ 3 & 0\end{bmatrix}$, find a,d,e.

Solution : If two matrices are equal, then their corresponding elements must be same.

Therefore, a = 2, d = -1, e = 3

Example 2 : Find the value of a, b and c. Given that -

$\begin{bmatrix}a & b\\ 8 & c\end{bmatrix}$ = $\begin{bmatrix}7 & 9\\ 8 & 3\end{bmatrix}$

Solution: Given

$\begin{bmatrix}a & b\\ 8 & c\end{bmatrix}$ = $\begin{bmatrix}7 & 9\\ 8 & 3\end{bmatrix}$

Since the matrices are equal the corresponding values are the same. Therefore

a = 7 b = 9 and c = 3.

Example 3 : Are the given matrices equal

$\begin{bmatrix}2 & 4\\ 6 & 8\end{bmatrix}$ = $\frac{1}{2}$ $\begin{bmatrix}4 & 8\\ 12 & 16\end{bmatrix}$

Solution:

$\begin{bmatrix}2 & 4\\ 6 & 8\end{bmatrix}$ = $\frac{1}{2}$ $\begin{bmatrix}4 & 8\\ 12 & 16\end{bmatrix}$

$\begin{bmatrix}2 & 4\\ 6 & 8\end{bmatrix}$ = $\begin{bmatrix}\frac{4}{2} & \frac{8}{2}\\ \frac{12}{2} & \frac{16}{2}\end{bmatrix}$


$\begin{bmatrix}2 & 4\\ 6 & 8\end{bmatrix}$ = $\begin{bmatrix}2 & 4\\ 6 & 8\end{bmatrix}$

Every elements are equal so they are equal matrices

Example 4 : $\begin{bmatrix}a-b & 3\\ 2 & a+b\end{bmatrix}$ = $\begin{bmatrix}2 & 3\\ 2 & 6\end{bmatrix}$ Find the value of a and b?

Solution: Given $\begin{bmatrix}a-b & 3\\ 2 & a+b\end{bmatrix}$ = $\begin{bmatrix}2 & 3\\ 2 & 6\end{bmatrix}$
a - b = 2
a + b = 6

---------------------------- Adding

2a = 8
---------------------------

a = $\frac{8}{2}$ = 4

Substituting in second equation.
a - b = 2
4 - b = 2
-b = 2 - 4
-b = -2
b = 2

Practice Problems

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Practice with the following problems.

Problem 1 : Determine the values of p, q and r from the relation below.

$\begin{bmatrix}4 & 0 & 6\\-2 & 3 & 1\end{bmatrix}$ = $\begin{bmatrix}p & 0 & 6\\ q+4 & \frac{r}{3} & 1\end{bmatrix}$

Answer : p = 4, q = - 6, r = 9.
Problem 2 : Find the value of a, b and c if following two matrices are equal.

$\begin{bmatrix}a+3 & -1\\ 4 & 5\end{bmatrix} = \begin{bmatrix}6 & b\\ c-3 & 5\end{bmatrix}$

Answer : a = 3, b = −1, c = 7

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