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Equality of Matrices
Equality of matrices is a concept when two or more matrices are equal. Matrices are said to be equal if they have the same number of rows, the same number of columns and their corresponding elements are equal. Equality of matrices does not hold of either of the mentioned properties does not hold. This implies if the order of the matrices is not equal or at least one pair of the corresponding elements is not equal, then the two matrices are said to be unequal.
Further, in this article, we will understand the definition of equality of matrices and the conditions that are required for matrix equality. We will also learn to solve for the equality of matrices with the help of examples for a better understanding.
1.  What is Equality of Matrices? 
2.  Equality of Matrices Definition 
3.  Solving for Matrices with Equality 
4.  FAQs on Equality of Matrices 
What is Equality of Matrices?
Equality of matrices is a mathematical concept of matrices where two or more matrices are said to be equal if they have the same dimensions and all corresponding elements of the matrices are equal. Equality of matrices is a concept that is true for any kind of matrix (rectangular and square). Equal matrices have the same number of rows and columns. Given below is an example of the equality of matrices A and B:
Equality of Matrices Definition
Two matrices A = [a_{ij}] and B = [b_{ij}] are said to be equal if and only if A and B have the same order, i.e., A and B have the same number of rows and the same number of columns, and corresponding elements of A and B are equal, i.e., a_{ij} = b_{ij} for all i and j.
The above statement is the main definition for the equality of matrices. Matrix Equality does not hold if either of the abovementioned properties is not true for the matrices A and B. In this case, matrices A and B are said to be unequal.
Conditions for Matrix Equality
Now, let us go through the main conditions required for the equality of matrices to be true. Given below are the three conditions required for matrix equality for matrices A = [a_{ij}]_{m×n} and B = [b_{ij}]_{p×q} :
 Matrices A and B have the same number of rows, i.e., m = p
 Matrices A and B have the same number of columns, i.e., n = q
 Corresponding elements of A and B are equal, i.e., a_{ij} = b_{ij} for all i and j.
Let us consider an example of two matrices. Consider row matrices A = [1 2 x]_{1×3} and B = [y 2 7 ]_{1×3} Now both the matrices A and B have the same number of rows that is 1 and the same number of columns that is 3. Hence they have the same dimensions. Now, if the equality of matrices holds for these two matrices, then their corresponding elements are equal, that is, 1 = y, 2 = 2 and x = 7.
Solving for Matrices with Equality
We have understood the meaning of equality of matrices. Now, let us understand how to solve equal matrices. We will consider two equal matrices:
\(\left[\begin{array}{ccc} 3x+4y & x2y & 6 \\ a+b & 3 & 2ab \end{array}\right] = \left[\begin{array}{ccc} 2 & 4 & 6 \\ 5 & 3 & 5 \end{array}\right]\)
The order of the above two matrices is equal, thus the equality of matrices hold if and only if the corresponding elements are also equal. Therefore, we have:
⇒ 3x + 4y = 2,  (1)
x  2y = 4,  (2)
a + b = 5,  (3)
2a  b = 5  (4)
Solving equations (1) and (2), we have
x = 2y + 4 [From (2)]
Substituting the above in (1),
3(2y + 4) + 4y = 2
⇒ 6y + 12 + 4y = 2
⇒ 10y = 2  12
⇒ 10y = 10
⇒ y = 1
⇒ x = 2(1) + 4
= 2 + 4
= 2
Similarly, solving equations (3) and (4), we have a = 0 and b = 5.
Important Notes on Equality of Matrices
 Equal matrices have the same number of rows and columns.
 Matrix Equality holds if and only if the matrices have the same order and the corresponding elements are equal.
Related Topics on Equality of Matrices
Equality of Matrices Examples

Example 1: Check if the matrices \(\left[\begin{array}{ccc} 2 & 3 & 6 \\ 4 & 5 & 7 \\ 1 & 3 & 12 \end{array}\right]\) and \(\left[\begin{array}{ccc} 1 & 3 & 6 \\ 4 & 5 & 7 \\ 1 & 3 & 12 \end{array}\right]\) are equal using equality of matrices definition.
Solution: The given two matrices have the same order, i.e., they have the same number of rows and columns. So, the first condition for matrix equality is satisfied.
Now, we will check for the corresponding elements of the matrices. As we can see the element in the first row and the first column is 2 in the first matrix and 1 in the second matrix and 2 is not equal to 1.
This contradicts the second condition of equality of matrices.
Therefore, the matrices are not equal.
Answer: Hence the given two matrices are not equal.

Example 2: Two matrices A = [a+b 6 8 2x 3b] and B = [3 6 8 14 9] are equal. Find the values of a, b , x.
Solution: Since A and B are given to be equal matrices, therefore their corresponding elements are also equal. We have
a + b = 3, 2x = 14, 3b = 9
⇒ x = 14/2 = 7, b = 9/3 = 3
⇒ a + 3 = 3 [From b = 3]
⇒ a = 0
⇒ a = 0, b = 3, x = 7
Answer: Hence, a = 0, b = 3, x = 7
FAQs on Equality of Matrices
What is Equality of Matrices in Math?
Equality of matrices is a mathematical concept of matrices where two or more matrices are said to be equal if they have the same dimensions and all corresponding elements of the matrices are equal.
What are the Conditions for Equality of Matrices?
Given below are the three conditions required for matrix equality for matrices A = [a_{ij}]_{m×n} and B = [b_{ij}]_{p×q} :
 Matrices A and B have the same number of rows, i.e., m = p
 Matrices A and B have the same number of columns, i.e., n = q
 Corresponding elements of A and B are equal, i.e., a_{ij} = b_{ij} for all i and j.
How to Solve Equality of Matrices?
Equal matrices can be solved by comparing the corresponding elements of the two equal matrices and determining the values of the unknown variables, if any.
How Do You Prove Two Matrices are Equal?
Equality of matrices can be proved by proving that the matrices have the same number of rows and the same number of columns and their corresponding elements are equal.
How to Check if the Two Matrices are Equal?
We check for matrix equality if the matrices have the same number of rows and columns. We also check if the corresponding elements of the matrices are equal.
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