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.
