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Question Video: Finding the Transpose of a Matrix Mathematics

Given the matrix 𝐴 = [−8, 4, 3 and 4, 1, −1], find (𝐴^(𝑇))^(𝑇).

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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.

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