Question Video: The Properties of Multiplication and Transpose of a Matrix Mathematics • 10th Grade

Given that 𝐵𝐴 = [8, −4 and −7, −5], what is 𝐴^𝑇 𝐵^𝑇?

02:28

Video Transcript

Given that 𝐵 multiplied by 𝐴 is equal to the two-by-two matrix eight, negative four, negative seven, negative five, what is the transpose of matrix 𝐴 multiplied by the transpose of matrix 𝐵?

In this question, we’re given that matrix 𝐵 multiplied by matrix 𝐴 is a given two-by-two matrix. And we need to use this to determine the transpose of matrix 𝐴 multiplied by the transpose of matrix 𝐵. To do this, we’re going to need to find a relationship between 𝐵 multiplied by 𝐴 and the transpose of 𝐴 multiplied by the transpose of 𝐵. And we can do this by recalling how we distribute the transpose over matrix multiplication.

We can recall if matrix 𝐵 multiplied by matrix 𝐴 is well defined, then we can distribute the transpose over this matrix multiplication. All we need to do is take the transpose of each matrix and switch the order of the multiplication. The transpose of matrix 𝐵 times 𝐴 is equal to the transpose of matrix 𝐴 multiplied by the transpose of matrix 𝐵. And we can see the right-hand side of this equation is what we’re asked to find in the question and we’re given the matrix 𝐵 multiplied by 𝐴.

So we can substitute this matrix into the equation. We get the transpose of matrix 𝐴 multiplied by the transpose of matrix 𝐵 is equal to the transpose of the two-by-two matrix eight, negative four, negative seven, negative five.

Now all we need to do is evaluate the transpose of this two-by-two matrix. And we can do this by recalling to take the transpose of a matrix, we write the corresponding rows of this matrix as the corresponding columns of the new matrix. And we can do this row by row. Let’s start with the first row of this matrix, which is the row eight, negative four. We need to write this row as the first column in the transpose. So the first column is eight, negative four.

We can then do the same to the second row. The second column of the transpose of this matrix will be negative seven, negative five. And since this was the last row, this gives us the transpose of this matrix. It’s the two-by-two matrix eight, negative seven, negative four, negative five. And this is our final answer. We were able to show if 𝐵 times 𝐴 is the two-by-two matrix eight, negative four, negative seven, negative five, then the transpose of 𝐴 times the transpose of 𝐵 is the two-by-two matrix eight, negative seven, negative four, negative five.

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