Question

If A is a square matrix with $\left| A \right|=6$. Find $\left| A{A}' \right|$?

Answer 1

This type of problem is based on the concept of matrix and determinant. Here, we find that the determinant of matrix A is 6. We know that $\left| AB \right|=\left| A \right|\left| B \right|$, where A and B are matrices. Using this, we get $\left| A{A}' \right|=\left| A \right|\left| {{A}'} \right|$, where ${A}'$ is the transpose of A. Since, the determinant of transpose of a matrix is equal to determinant of the same matrix, that is $\left| {{A}'} \right|=\left| A \right|$, we get $\left| A{A}' \right|=\left| A \right|\left| A \right|$. From the question $\left| A \right|=6$ and thus, $\left| A{A}' \right|=6\times 6$. Do necessary calculations to get the final required answer.

Complete step by step solution:
According to the question, we are asked to find $\left| A{A}' \right|$ for a square matrix A.
We have been given that $\left| A \right|=6$. ---------------(1)
That is the determinant of a square matrix A is equal to 6.
We know that for two square matrices A and B,
$\left| AB \right|=\left| A \right|\left| B \right|$
Therefore, we get
$\left| A{A}' \right|=\left| A \right|\left| {{A}'} \right|$ --------------(2)
We know that ${A}'$ is the transpose of matrix A.
Using the fact that determinant of the transpose of a matrix is equal to determinant of that matrix, we get
$\left| {{A}'} \right|=\left| A \right|$
On substituting the above result in equation (2), we get
$\left| A{A}' \right|=\left| A \right|\left| A \right|$
On further simplification, we get
$\left| A{A}' \right|={{\left| A \right|}^{2}}$
But we have been given in the question that $\left| A \right|=6$.
On substituting this value in the above equation, we get
$\left| A{A}' \right|={{6}^{2}}$
We know that the square of 6 is 36.
Therefore, we get
$\left| A{A}' \right|=36$

Hence, the value of $\left| A{A}' \right|$ for $\left| A \right|=6$ is 36.

Note: Whenever we get such a type of problem, we have to use the property of determinants to solve it. We should not add the determinant of A with the determinant of transpose of A which will lead to a wrong answer. Avoid calculation mistakes to get the accurate answer. Similarly, we can solve for three by three matrices also.