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

Last updated date: 29th Feb 2024
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Hint: 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.