Video Transcript
If 𝐴 is a matrix of order three by three such that the determinant of 𝐴 is equal to two, find the determinant of the transpose of 𝐴.
In this question, we’re given a three-by-three matrix 𝐴, and we’re told the determinant of this three-by-three matrix is two. We need to use this information to find the value of the determinant of 𝐴 transpose. The first thing worth noting is the transpose of a matrix replaces its rows with its columns. So since we’re told that 𝐴 is a three-by-three matrix, the transpose of 𝐴 is also a three-by-three matrix. This means it’s a square matrix, so we can evaluate its determinant.
In particular, we can evaluate this determinant by recalling if 𝐴 is any square matrix, then the determinant of 𝐴 is equal to the determinant of 𝐴 transpose. Therefore, the determinant of the transpose of 𝐴 is equal to the determinant of 𝐴. And we’re told in the question this is equal to two, which means we’ve shown if 𝐴 is a matrix of order three by three such that the determinant of 𝐴 is equal to two, then the determinant of the transpose of 𝐴 is also equal to two.
