Question Video: Applying Operations on Matrices to Find the Values of Unknowns Involving Scalar Multiplication

Video Transcript

Given that 𝑥 multiplied by the matrix negative three, negative eight, negative eight plus 𝑦 multiplied by the matrix zero, zero, seven minus 𝑧 multiplied by the matrix zero, four, negative one is equal to the matrix negative 12, negative 28, negative 19. Find the values of 𝑥, 𝑦, and 𝑧.

Well, the first thing we do is multiply the matrices by 𝑥, 𝑦, and 𝑧. So, first of all, we’re gonna have negative three 𝑥, negative eight 𝑥, negative eight 𝑥. Then, plus the matrix zero, zero, seven 𝑦. And we get that because 𝑦 multiplied by zero is just zero. 𝑦 multiplied by the other zero is just zero. And then, 𝑦 multiplied by seven is seven 𝑦.

And then, we’re gonna have minus, then the matrix zero, four 𝑧, negative 𝑧. And again, that’s because when we multiply 𝑧 by zero gives us zero, 𝑧 by four gives us four 𝑧, and then 𝑧 by negative one is the same as negative one 𝑧. But we just write that as a negative 𝑧. Then, this is all equal to the matrix negative 12, negative 28, negative 19.

Okay, great. So, now what do we need to do if we want to evaluate and find the values of 𝑥, 𝑦, and 𝑧? Well, what we can do is we can set up some simultaneous equations using the corresponding components that we have. So, first of all, we’re gonna start with our top component. And we can see from our top component, we’ve got negative three 𝑥, then plus zero minus zero so we’re gonna ignore those, is equal to negative 12. And I’m gonna call this equation one.

So then, we’re gonna look at the next components. From our next components, we can form the equation negative eight 𝑥, then we’ve got plus zero so we can ignore this, minus four 𝑧 is equal to negative 28. So, we’re gonna call this equation two.

Then, from the bottom components, we can form our third equation. And that is negative eight 𝑥 plus seven 𝑦 minus negative 𝑧 is equal to negative 19. But we know if we subtract a negative, it’s gonna be positive. So, we can rewrite this as negative eight 𝑥 plus seven 𝑦 plus 𝑧 is equal to negative 19. Okay, great. So, now we can use these equations to find 𝑥, 𝑦, and 𝑧. Well, I’m gonna start with equation one. And that’s because this is this most useful to start with because we only have one variable because it just contains 𝑥.

So, we have negative three 𝑥 equals negative 12. So, what we can do is divide each side of the equation by negative three. And that’s because if we got negative three 𝑥, well you want one 𝑥 or just 𝑥, divide it by negative three and we get 𝑥. And when we do that to the other side of the equation — cause whatever we do to one side, we must do to the other — we get four because negative 12 divided by negative three is four because a negative divided by a negative is a positive.

So, now that’s enabled us to find 𝑥. So, what we want to do is find the next variable, and what we’re gonna do is find 𝑧 next. And we’re gonna do that by substituting 𝑥 equals four into equation two. And when we do that, we get negative eight multiplied by four, and that’s because 𝑥 was equal to four, minus four 𝑧 is equal to negative 28. Well, negative eight multiplied by four is negative 32. So, we’ve got negative 32 minus four 𝑧 equals negative 28.

So, we can add 32 to each side of the equation. And when we do that, we get negative four 𝑧 is equal to four. So therefore, if we divide both sides of the equation by negative four to find our single 𝑧, we get 𝑧 is equal to negative one. And that’s cause if we have four divided by negative four, we get negative one because a positive divided by a negative is a negative.

So great, we found 𝑥 and 𝑧. So, now let’s move on to the equation three to find 𝑦. And we can do that by substituting in 𝑥 equals four and 𝑧 equals negative one. So, when we do that, we get negative eight multiplied by four plus seven 𝑦 minus one is equal to negative 19. And that’s again because 𝑥 was equal to four and 𝑧 is equal to negative one. As we already stated earlier, negative eight multiplied by four is negative 32, so we can rewrite it as negative 32 plus seven 𝑦 minus one equals negative 19.

So then, if we simplify, we get seven 𝑦 minus 33 equals negative 19. And that’s cause negative 32 minus one is negative 33. And if we add 33 to each side of our equation, what we’re gonna get is seven 𝑦 is equal to 14. Then, we divide both sides of the equation by seven because we want to find our single 𝑦. And when we do that, we have 𝑦 is equal to two.

So, now we found the individual variables and we can say that the value of 𝑥 is four, the value of 𝑦 is two, and the value of 𝑧 is negative one. And these rule, given that 𝑥 multiplied by the matrix negative three, negative eight, negative eight plus 𝑦 multiplied by the matrix zero, zero, seven minus 𝑧 multiplied by the matrix zero, four, negative one is equal to the matrix negative 12, negative 28, and negative 19.