### Video Transcript

Given that the matrix seven, negative two, negative five, two, 10, negative two, five, negative eight, five plus the matrix 11, 11, nine, zero, 11, five, six, one, two is equal to the matrix 𝑥, nine, 𝑦, two, 𝑧, three, 𝑎, negative seven, 𝑏, find the values of 𝑥, 𝑦, 𝑧, 𝑎, and 𝑏.

What we’ve been given in this question is a matrix equation. The two matrices on the left-hand side sum together to give the matrix on the right-hand side. 𝑥, 𝑦, 𝑧, 𝑎, and 𝑏 represent five unknown elements in the answer matrix. We know that in order to add two matrices together, they must be of the same order, and the answer is a matrix also of the same order. We see here that the two matrices we’re adding each have three rows and three columns, and so does the answer matrix. So, they’re all of order three by three.

We should recall that when we add two matrices, we add corresponding elements. That means we add elements that are in the same row and same column to give the elements in the same position in the answer matrix. So, in the second row and first column, we have two plus zero, which is equal to two. We can therefore work out the values of our five unknowns by adding the elements in the same position as them in the other two matrices.

For example, the unknown 𝑥 is in the first row and first column, so it’s the result of adding the elements in the first row and first column of the other two matrices. 𝑥 is therefore equal to seven plus 11, which is equal to 18. 𝑦 is the element in the first row and third column of the answer matrix. So, we find the value of 𝑦 by adding together the elements in this position in the other two matrices. 𝑦 is equal to negative five plus nine, which is equal to four. To find the value of 𝑧, we sum the elements in the second row and second column giving 10 plus 11, which is equal to 21.

To find 𝑎, we sum the elements in the third row and first column of the two matrices giving five plus six, which is equal to 11. And finally, for 𝑏, we sum the elements that are in the third row and third column of the two matrices, giving five plus two, which is equal to seven.

We found all five unknowns. 𝑥 is equal to 18, 𝑦 is equal to four, 𝑧 is equal to 21, 𝑎 is equal to 11, and 𝑏 is equal to seven.