Question Video: Finding Unknown Elements of a Matrix Using Equality of Matrices Mathematics

Video Transcript

Given that the matrix negative four, three, 𝑥, negative seven is equal to the matrix negative four, three, eight, 𝑦 minus six, find the values of 𝑥 and 𝑦.

The question gives us two matrices which we are told are equal. We need to use this information to find the values of 𝑥 and 𝑦. Remember, for two matrices to be equal, entries in the same row and column must be identical. So, for example, we must have both entries in the first row and first column equal. In this case, they’re both equal to negative four. But this doesn’t really help us find the values of 𝑥 or 𝑦. However, what happens if we look at the values in row two column one? Remember, these must be equal. In our first matrix, the value in row two column one is 𝑥. And in our second matrix, the value in row two column one is eight. So for our matrices to be equal, these two entries must be equal. In other words, we must have 𝑥 is equal to eight.

We’ll want to do something similar to help us find the value of 𝑦. We can see the only place 𝑦 appears is in our second matrix in row two column two. And for these two matrices to be equal, they must have the same value in row two column two. So we can just equate the entries in row two column two for both of these matrices. In other words, we must have negative seven is equal to 𝑦 minus six. And we can then just solve this equation for 𝑦. We’ll add six to both sides of the equation. And we see that this gives us that 𝑦 is equal to negative one.

One thing that’s often worth doing in situations like this is substituting our value of 𝑦 back into our matrix. Remember, when we do this, we should get the entry of negative seven. So we’ll substitute 𝑦 is negative one into the expression in row two column two in our second matrix. This gives us negative one minus six. And we can evaluate this, and we get negative seven just as we expected. This just helps us check that our answer was correct. Therefore, given the matrix negative four, three, 𝑥, negative seven is equal to the matrix negative four, three, eight, 𝑦 minus six, we were able to show that the value of 𝑥 must be eight and the value of 𝑦 must be negative one.