### Video Transcript

Given the matrix 𝐴 equals the
two-by-three matrix negative eight, four, three, four, one, negative one, find the
transpose of 𝐴 transpose.

In this question, we are given a
two-by-three matrix 𝐴 and asked to find the transpose of the transpose of this
matrix. We can do this in two ways. First, we recall that we find the
transpose of a matrix by switching the rows with the corresponding columns of the
matrix. We can use this to find the
transpose of matrix 𝐴. We can start by writing the first
row of matrix 𝐴 as the first column in its transpose. This gives us a first column of
negative eight, four, three. We can follow the same process for
the second row of 𝐴. We write this as the second column
in the transpose of 𝐴 to obtain a second column of four, one, negative one.

We want to find the transpose of 𝐴
transpose, so we need to apply this process once again to this new matrix 𝐴
transpose. We start by writing the first row
of this matrix as the first column in its transpose. This gives us a first column of
negative eight, four. We then write the second row of
this matrix as the second column of its transpose. We obtain a second column of four,
one. We follow this process one final
time by writing the final row of the matrix as the final column of the new
matrix. The final column is three, negative
one. Therefore, we have shown that the
transpose of 𝐴 transpose is the two-by-three matrix negative eight, four, three,
four, one, negative one.

However, this is not the only way
we can answer this question. We can note that the transpose of
𝐴 transpose is actually equal to 𝐴. We can show why this is true by
noting that taking the transpose of 𝐴 transpose will switch the rows with the
columns and then switch them back. So for any matrix 𝑀, taking the
transpose of 𝑀 transpose will leave the matrix unchanged. We can apply this result with
matrix 𝐴 instead of 𝑀 to get that the transpose of 𝐴 transpose is equal to 𝐴,
the two-by-three matrix negative eight, four, three, four, one, negative one.