### Video Transcript

If 𝐴 is a matrix of order three by
three such that the determinant of 𝐴 is two, find the determinant of two 𝐴.

We recall the property of
determinants which says that for any square matrix 𝐴 of order 𝑛 by 𝑛 and scalar
𝑘, the determinant of 𝑘𝐴 equals 𝑘 to the power of 𝑛 times the determinant of
𝐴. In this question, we are told
matrix 𝐴 has order three by three, which means it has three rows and three
columns. We are also told that the
determinant of matrix 𝐴 is two.

To answer this question, we need to
find the determinant of two 𝐴. Since 𝐴 is a three-by-three
matrix, 𝑛 equals three. We will be using a scalar 𝑘 of
two. Therefore, according to the
property of determinants, by using 𝑛 equals three and 𝑘 equals two, we find that
the determinant of two 𝐴 is two to the power of three times the determinant of
𝐴. That is eight times the determinant
of 𝐴. And the determinant of 𝐴 is
two. Therefore, the determinant of two
𝐴 equals the product of eight and two, which is 16.

In conclusion, given a
three-by-three matrix 𝐴 with a determinant of two, the determinant of two 𝐴 is
16.