Question

If $A$ is $2 \times 3$ matrix and $AB$ is a $2 \times 5$ matrix, then $B$ must be a
(A) $3 \times 5$matrix
(B) $5 \times 3$matrix
(C) $3 \times 2$matrix
(D) $5 \times 2$matrix

Answer 1

Hint: The matrix multiplication should be possible only if $A$ =$'m \times n'$ matrix, then $B$ should be $'n \times b'$matrix and thus $AB$ becomes $'m \times b'$matrix.

Complete step-by-step answer:
 $A$=$2 \times 3$matrix
& $AB$=$2 \times 5$matrix
We know that if $A$ =$'m \times n'$ matrix, then $B$ should be $'n \times b'$matrix, so that the matrix multiplication is possible and $AB = 'm \times b'$matrix.
In other words, Multiplication of matrices follows-
${\left[ A \right]_{m \times n}}{\left[ B \right]_{n \times b}} = {\left[ {AB} \right]_{m \times b}}$
Given $m = 2$, $n = 3$, $b = 5$
So, $B$ must be a $3 \times 5$ matrix.
Hence, option (A) is the correct answer.

Note: Some facts related to the multiplication of matrices are given below:
AB is possible, only if-
Number of columns in matrix A = Number of rows in matrix B
Multiplication of matrices follows-
${\left[ A \right]_{m \times n}}{\left[ B \right]_{n \times b}} = {\left[ {AB} \right]_{m \times b}}$