Video: Using Properties of Equal Matrices to Find the Values of Unknowns

Video Transcript

Given that the matrix negative 216, three, zero, negative six 𝑧 plus 𝑦 is equal to the matrix 𝑙 cubed, 𝑥 squared plus two, negative nine 𝑦, 60, find the values of 𝑥, 𝑦, 𝑧, and 𝑙.

So to work out the values of 𝑥, 𝑦, 𝑧, and 𝑙, what we can do is we can equate each of the corresponding components from our matrices. And we’re gonna start with negative 216 is equal to 𝑙 cubed. Well, if negative 216 is equal to 𝑙 cubed and we want to find 𝑙, then what we need to do is take the cube root of both sides of our equation. So when we do that, we’re gonna get negative six is equal to 𝑙. And we get that because negative six multiplied by negative six gives us positive 36. And then positive 36 multiplied by negative six is gonna give us negative 216. And that’s because a positive multiplied by a negative is negative. Okay, great, that’s one of our values.

So now, what I’m gonna do is I’m gonna move on to our next components. So our next corresponding components are three and 𝑥 squared plus two. So therefore, I can say that three is equal to 𝑥 squared plus two. So then, if we subtract two from each side of the equation, we’re gonna be left with one is equal to 𝑥 squared. And now, because we’ve got 𝑥 squared and we wanna find 𝑥, what we’re gonna do is take the square root of both sides of the equation. And when we do that, we’re left with positive or negative one is equal to 𝑥. And we get both positive or negative result because positive one multiplied by positive one would give us one. And negative one multiplied by negative one would also give us one. So therefore, there are two possible answers. So great, that’s 𝑥 found.

So now, we move on to our third pair of components. We’ve got zero and negative nine 𝑦. Well, if we’ve got zero is equal to negative nine 𝑦. And we can clearly see that zero is equal to 𝑦. And we could’ve worked it out by dividing each side by negative nine. And then, we would’ve got zero divided by negative nine, which is zero. Because, in fact, zero divided by any positive or negative number is zero. So great, we’ve now got our third value.

So now, finally, we move to our last components. And these are negative six 𝑧 plus 𝑦 and 60. So we get negative six 𝑧 plus 𝑦 is equal to 60. Well, we know that negative six 𝑧 is gonna be equal to 60. And that’s because 𝑦 is equal to zero. So negative six 𝑧 plus zero is just negative six 𝑧. So then, what we do is we divide each side of the equation by negative six. And when I do that, we’re left with 𝑧 is equal to negative 10.

So therefore, we can say that the final values of 𝑥, 𝑦, 𝑧, and 𝑙 are: 𝑙 is equal to negative six, 𝑥 is equal to positive or negative one, 𝑦 is equal zero, and 𝑧 is equal to negative 10.