Question

If $\left( {123} \right)B = \left( {34} \right)$ then the order of the matrix $B$ is
$\left( a \right){\text{ 3}} \times {\text{1}}$
$\left( b \right){\text{ 3}} \times 2$
$\left( c \right){\text{ 2}} \times 4$
$\left( d \right){\text{ 5}} \times 2$

Answer 1

This type of problem will be solved by using the concept of multiplication of matrices and it is given by ${A_{p \times q}} \times {B_{q \times r}} = {C_{p \times r}}$ . So by comparing this formula we will be able to find the answer easily which will be the matrix $B$.

Formula used:
Multiplication of matrices is given by
${A_{p \times q}} \times {B_{q \times r}} = {C_{p \times r}}$
Here,
$A,B,C$ are the matrices of any order of the matrix
And $p \times q,q \times r,p \times r$ will be the order of the matrix.

Complete step by step solution:
So it is given that $\left( {123} \right)B = \left( {34} \right)$ and we have to find the order of it.
So for solving it the multiplication of matrices is used.
As we can see $\left( {123} \right)$ is the matrix having the order of $1 \times 3$ and also for the other matrix which is $\left( {34} \right)$ having the order of $1 \times 2$.
Therefore, for the matrix $\left( {123} \right)$ having the order of the matrix is $1 \times 3$
So, ${A_{p \times q}} = {\left( {123} \right)_{1 \times 3}}$ and here, $p = 1,q = 3$.
Also, for the matrix $\left( {34} \right)$ having the order of the matrix is $1 \times 2$
So, ${C_{p \times r}} = {\left( {34} \right)_{1 \times 2}}$ and here, $p = 1,r = 2$.
So now on comparing the values with the formula we get
$\Rightarrow {(123)_{1 \times 3}} \times {B_{q \times r}} = {(34)_{1 \times 2}}$ we get $p = 1,q = 3,r = 2$
Therefore, the order of the matrix $B$ will be $3 \times 2$.

Hence, the option $\left( b \right)$ is correct.

Note:
We should always keep in mind while solving it that the row of one matrix and column of the other matrix which is involved in the multiplication should be the same. So by using it we easily choose the answer from the option or also it will be used for the rechecking, whether we are correct or not while solving the problem related to addition or the multiplication of the matrices.