Question

If $AB = C$, then which of the following options is correct regarding matrices $A$, $B$, and $C$?
A. ${A_{2 \times 3}},{B_{3 \times 2}},{C_{2 \times 3}}$
B. ${A_{3 \times 2}},{B_{2 \times 3}},{C_{3 \times 2}}$
C. ${A_{3 \times 3}},{B_{2 \times 3}},{C_{3 \times 3}}$
D. ${A_{3 \times 2}},{B_{2 \times 3}},{C_{3 \times 3}}$

Answer 1

Hint: Consider a matrix, $A$ of order $l \times m$ and another matrix, $B$ of order $m \times n$ . Let’s say that matrix $C$ is formed as a result of the product of matrices $A$ and $B$, that is, $C = AB$. Then, the order of $C$ will be $l \times n$.

Complete step by step Solution:
We are given three matrices, $A$ , $B$ and $C$ such that $AB = C$ .
Now, we know that the order of a product of two matrices is the number of rows of the first matrix by the number of columns of the second matrix.
Let us now consider each option and check if they satisfy the above condition.
For option A:
Order of Matrix $A = 2 \times 3$
Order of Matrix $B = 3 \times 2$
They can be multiplied. Therefore, order of Matrix $C = 2 \times 2$
Given order of Matrix $C = 2 \times 3$
Hence, this is an incorrect option.
For option B:
Order of Matrix $A = 3 \times 2$
Order of Matrix $B = 2 \times 3$
They can be multiplied. Therefore, order of Matrix $C = 3 \times 3$
Given order of Matrix $C = 3 \times 2$
Hence, this is an incorrect option.
For option C:
Order of Matrix $A = 3 \times 3$
Order of Matrix $B = 2 \times 3$
They cannot be multiplied.
Hence, this is an incorrect option.
For option D:
Order of Matrix $A = 3 \times 2$
Order of Matrix $B = 2 \times 3$
They can be multiplied. Therefore, order of Matrix $C = 3 \times 3$
Given order of Matrix $C = 3 \times 3$
Hence, this is the correct option.
Hence, the correct option is (D).

Note:Matrix multiplication of two matrices is only possible when the number of columns of the first matrix is equal to the number of rows of the second matrix. This means that a matrix $A$ of order $\left( {j \times k} \right)$ and another matrix $B$ of order $\left( {l \times m} \right)$ can only be multiplied when $k = l$ .