Question

If $A$ is a $3\times 3$ matrix such that $\left| A \right|=8$ then $\left| 3A \right|$ equals.
A) $8$
B) $24$
C) $72$
D) $216$

Answer 1

In this question we have been asked to find the determinant of matrix $3A$ when the determinant of $A$ is given as $8$ . For answering this question we will use the concept of matrices that states that $\left| xA \right|={{x}^{n}}\left| A \right|$ this expression is valid where $x$ is a scalar and $A$ is a matrix and $n$ is the order of the matrix.

Complete step by step solution:
Now considering from the question we need to find the determinant of matrix $3A$ when the determinant of $A$ is given as $8$ .
For answering this question we will use the concept of matrices that states that when the matrix $A$ is multiplied by a scalar $x$ then the determinant of the resultant matrix will be given by the product of the determinant of the matrix $A$ and the scalar $A$ which is mathematically given as $\left| xA \right|={{x}^{n}}\left| A \right|$ .
Now as we have $\left| A \right|=8$ and order of $A$ is $3$ from the question then we will have
$\begin{aligned} & \Rightarrow \left| 3A \right|={{3}^{3}}\times \left| A \right| \\\ & \Rightarrow {{3}^{3}}\times 8=216 \\\ \end{aligned}$
Hence we can conclude that the determinant of $3A$ is given as $216$ when it is given as $\left| A \right|=8$ and the order of $A$ is $3$ .

So, the correct answer is “Option D”.

Note: While answering questions of this type we should be sure with our matrices and determinants concepts. This question can be answered easily and in a short span of time and very few mistakes are possible in it. We have many other determinant properties similarly for example there is property given as “The transpose of a matrix has the same determinant as the determinant of the matrix”.