Given that the matrix 𝐴 equals negative two, six, negative six, one, eight, four, find 𝐴 transpose.
Just a quick reminder about this notation, first of all, when we see a matrix and then a superscript capital 𝑇, this means we’re being asked to find the transpose of the matrix 𝐴. That’s the matrix we get when we swap the rows and columns of the matrix 𝐴 around.
Looking carefully at our matrix 𝐴, then, we can see that it has two rows and three columns. The matrix 𝐴 transpose, then, will have three rows and two columns. The number of rows in matrix 𝐴 is the number of columns in its transpose. And the number of columns in matrix 𝐴 is the number of rows in its transpose.
The first row in matrix 𝐴 — so that’s negative two, six, negative six — becomes the first column in its transpose matrix. So, we can fill in this first column. The second row of matrix 𝐴 — so that’s one, eight, four — will become the second column in our matrix 𝐴 transpose. So, by swapping the rows and columns of matrix 𝐴 around, we’ve found its transpose matrix. 𝐴 transpose is equal to the matrix negative two, one, six, eight, negative six, four.
We can also look at individual elements of these two matrices. For example, the element that was in the first row and second column of matrix 𝐴 — so that’s six — is now in the second row and first column of the transpose matrix. Their positioning of rows and columns has been swapped around. Once again, our matrix 𝐴 transpose is the three-by-two matrix negative two, one, six, eight, negative six, four.