Question

If $A$ is a square matrix of order $3$ and $\left| A \right| = 4$, find the value of $\left| {2A} \right|$.

Answer 1

We are given the order of the matrix and determinant of the same matrix. If a row or column is multiplied by a number $k$, the determinant is also multiplied by the same number. Since the order is three, there are three rows and columns. Using this we can find the determinant of $2A$.

Useful formula:
The determinant of the matrix $kA$, $\left| {kA} \right| = {k^n}\left| A \right|$, where $n$ is the order of the matrix.

Complete step by step solution:
Given that $A$ is a square matrix of order $3$.
A square matrix is a matrix with the same number of rows and columns. The order of the square matrix is the number of its rows (or columns).
Also determinant of $A$, $\left| A \right| = 4$
We have to find the determinant of the matrix $2A$.
If a row or column is multiplied by a number $k$, the determinant is also multiplied by the same number.
When we consider $kA$, every row of the matrix is multiplied by the number $k$.
So the determinant of the matrix $kA$, $\left| {kA} \right| = {k^n}\left| A \right|$, where $n$ is the order of the matrix.
Since the order of the matrix is given as $3$, we have
$\left| {2A} \right| = {2^3}\left| A \right|$
Substituting the determinant of the matrix $A$ we get,
$\left| {2A} \right| = {2^3} \times 4$
Simplifying we get,
$\left| {2A} \right| = 8 \times 4 = 32$

Therefore the answer is $32$.

Note:
In the question, it is said that the matrix is a square matrix. The determinant is defined only for square matrices. For a square matrix, every scalar multiple will also be a square matrix. So the determinant is defined for $2A$.