Solving Equality of Matrices

We have learned what is meant by the equality of matrices and also the conditions that are required for the equality of matrices. Now, let us solve the equality of matrices. For example, let us consider two matrices, A = [aij] and B = [bij]. Now, the two matrices A and B are said to be equal if and only if the order of both matrices is the same and also their corresponding elements are equal, i.e., a_ij = b_ij for all i and j.

First, let us consider two equal matrices A and B.

As the given matrices are equal, so A = B.

We can see that the order of the given matrices is equal, so the equality of matrices holds if and only if the corresponding elements are also equal.

So, 2a + 3b = 5 ⇢ (1)

a + b = 1 ⇢ (2)

a = 1 − b ⇢ (3)

Now, substitute the value of a = 1 − b in equation (1)

⇒ 2 (1 − b) + 3b = 5

⇒ 2 − 2b + 3b = 5

⇒ 2 + b = 5

⇒ b = 5 − 2 = 3

Now, substitute the value of b =3 in equation (3)

⇒ a = 1 − 3 = −2

Thus, the given matrices are said to be equal if a = −2 and b =3.

The equality of matrices is a mathematical concept where two or more matrices are equal when compared. Before learning the concept of equality of matrices, we need to know what a matrix is. A rectangle or square-shaped array of numbers or symbols organized in rows and columns to represent a mathematical object or one of its attributes is called a matrix in mathematics. The horizontal lines are said to be rows, while the vertical lines are said to be columns.  For example,  is a matrix with 3 rows and 3 columns. It can be called a “3 by 3” matrix and is a square matrix. On the other hand,  is a “2 by 3” matrix and is a rectangular matrix.