Question

If \[o\left( A \right) = 2 \times 3,o\left( B \right) = 3 \times 2\] and \[o\left( C \right) = 3 \times 3\], then which one of the following is not defined?
1. $CB + A'$
2. $BAC$
3. $C\left( {A + B'} \right)'$
4. $C\left( {A + B'} \right)$

Answer 1

Hint: Here, we are given the order of three matrices, and we have to check from all the options which matric is not defined. The first step is to find and write the order of the transpose of given matrices. Then, start finding the order of each option by breaking it into different parts. Also, use (Order of $AB = $Number of rows of $A \times $Number of columns of $B$) this formula to find the order.

Formula Used:
Order of $AB = $Number of rows of $A \times $Number of columns of $B$

Complete step by step Solution:
Let, $A,B,C$ are the three matrices whose orders are \[o\left( A \right) = 2 \times 3,o\left( B \right) = 3 \times 2\] and \[o\left( C \right) = 3 \times 3\]
It implies that the order of the transpose of following matrices will be
\[o\left( {A'} \right) = 3 \times 2,o\left( {B'} \right) = 2 \times 3,o\left( {C'} \right) = 3 \times 3\]
Now, let’s check for each of the options whether they are defined or not
1. $CB + A'$
Order of $CB = $order of $C \times $order of $B$
$o\left( {CB} \right) = 3 \times 2$
And \[o\left( {A'} \right) = 3 \times 2\]
Therefore, Matrix $CB + A'$ is defined.
2. $BAC$
Order of $BA = $order of $B \times $order of $A$
$o\left( {BA} \right) = 3 \times 3$ and \[o\left( C \right) = 3 \times 3\]
Therefore, Matrix $BAC$ is defined.
3. $C\left( {A + B'} \right)'$
$o\left( {A + B'} \right) = 2 \times 3$
$ \Rightarrow o\left( {A + B'} \right)' = 3 \times 2$ and \[o\left( C \right) = 3 \times 3\]
Therefore, Matrix $C\left( {A + B'} \right)'$ is defined.
4. $C\left( {A + B'} \right)$
$o\left( {A + B'} \right) = 2 \times 3$ and \[o\left( C \right) = 3 \times 3\]
Therefore, Matrix $C\left( {A + B'} \right)$ is not defined.


Hence, the correct option is 4.

Note:The key concept involved in solving this problem is a good knowledge of the matrix and its order. Students must remember that the order of the matrix determines the dimension of the matrix and the number of rows and columns in the matrix. The general representation of matrix order is ${A_{m \times n}}$, where $m$ is the number of rows and $n$ is the number of columns in the given matrix. In addition, the order of matrix multiplication answer ($m \times n$) gives the number of elements in the matrix.