Question Video: Finding Unknown Elements in a Matrix Using Equality of Matrices Mathematics • 10th Grade

Video Transcript

Consider the matrix 𝐴 is equal to negative 10𝑥, 𝑥 plus three 𝑦, two 𝑥 minus 𝑧 and the matrix 𝐵 is equal to negative 30, 27, 10. Given that the matrix 𝐴 is equal to the matrix 𝐵, determine the values of 𝑥, 𝑦, and 𝑧.

In this question, we’re given two matrices 𝐴 and 𝐵. And we can see that the entries in matrix 𝐴 are dependent on the three variables 𝑥, 𝑦, and 𝑧. In fact, we’re told that these two matrices are equal. We need to use this information to determine the values of 𝑥, 𝑦, and 𝑧. Remember, we say that a matrix 𝐴 is equal to a matrix 𝐵 if all of the entries in the same row and same column are equal. In fact, this also tells us that our matrices must have the same number of rows and columns.

In our case, matrix 𝐴 is negative 10𝑥, 𝑥 plus three 𝑦, two 𝑥 minus 𝑧 and matrix 𝐵 is negative 30, 27, 10. Since we’re told that these two matrices are equal, entries in the same row and same column must be equal. For example, the entry in the first row and first column of matrix 𝐴 is negative 10𝑥, and the entry in the first row and first column of matrix 𝐵 is negative 30. So for these two matrices to be equal, these two entries must be equal, so we get negative 10𝑥 must be equal to negative 30. And we can solve this equation for 𝑥. We just divide both sides by negative 10. And this gives us that our value of 𝑥 must be equal to three.

Let’s now move on to row one and column two of our two matrices. In matrix 𝐴, we can see that the entry in row one column two is equal to 𝑥 plus three 𝑦. And in matrix 𝐵, the entry in row one column two is equal to 27. So because these two matrices are equal, these two entries must be equal. This gives us that 𝑥 plus three 𝑦 must be equal to 27. Remember, we already showed that our value of 𝑥 must be equal to three, so we can substitute this into our equation. This gives us that three plus three 𝑦 must be equal to 27. We can then solve this equation for 𝑦. We’ll start by subtracting three from both sides of the equation. This gives us that three 𝑦 is equal to 24. Now to solve this equation for 𝑦, we’ll divide both sides of the equation through by three. And so we get that 𝑦 is equal to 24 divided by three, which is, of course, equal to eight.

We can do the same with the final entry in each of our two matrices, the entry in row one column three. In matrix 𝐴, this entry is two 𝑥 minus 𝑧. And in matrix 𝐵, this entry is 10. And remember, we’re told matrix 𝐴 is equal to matrix 𝐵, so we must have these two entries equal. So we get two 𝑥 minus 𝑧 is equal to 10. Remember, we already showed earlier that if our two matrices are equal, 𝑥 must be equal to three. So to help us find our value of 𝑧, we’ll substitute 𝑥 is equal to three into this equation. Substituting 𝑥 is equal to three, we get two times three minus 𝑧 is equal to 10. And we can just solve this equation for 𝑧. We add 𝑧 to both sides and then subtract 10 from both sides of the equation. This gives us that 𝑧 is equal to negative four. So this gives us that 𝑥 is equal to three, 𝑦 is equal to eight, and 𝑧 is equal to negative four.

But remember, it can be very useful to substitute these values back into our matrix to check that our answer is correct. So let’s substitute 𝑥 is equal to three, 𝑦 is equal to eight, and 𝑧 is equal to negative four into our matrix 𝐴. Substituting these values in, we get that our matrix 𝐴 is equal to negative 10 times three, three plus three times eight, two times three minus negative four. And if we evaluate each of these entries, we see we get negative 30, 27, 10. And each of these entries is exactly the same as we have in matrix 𝐵, so we know we have the right answer.

Therefore, if the matrix 𝐴 is equal to negative 10𝑥, 𝑥 plus three 𝑦, two 𝑥 minus 𝑧 and the matrix 𝐵 is equal to negative 30, 27, 10, then for 𝐴 to be equal to 𝐵, we must have that 𝑥 is equal to three, 𝑦 is equal to eight, and 𝑧 is equal to negative four.