Video Transcript
Determine the values of 𝑥, 𝑦, 𝑘, and 𝑙 that satisfy the given equation. 𝑥 multiplied by the matrix negative four, six, 10, 𝑘 plus 𝑦 multiplied by the matrix negative seven, 𝑙, zero, negative five plus four multiplied by the matrix three, negative one, 10, negative two is equal to 𝑂, where 𝑂 is the zero matrix of order two by two.
Well, the first thing we can see is that we actually want to carry out some scalar multiplication of our matrices because we can see that each matrix has a scalar. We’ve got 𝑥, 𝑦, and four. Now, to carry out scalar multiplication, what we do is multiply each element of the matrix by the scalar before the matrix.
When this is carried out with our first matrix, we’re gonna get negative four 𝑥, six 𝑥, 10𝑥, and 𝑘𝑥. Then, for our second matrix, we’re gonna have negative seven 𝑦, 𝑙𝑦, zero 𝑦, and negative five 𝑦. Then, the third matrix is 12, negative four, 40, negative eight. And then this is all equal to 𝑂, where we’re told that 𝑂 is the zero matrix of order two by two. So, what we can do is put this in instead.
So now, what we can do is, in fact, create four equations. And that’s because we know that negative four 𝑥 plus negative seven 𝑦 plus 12, because they’re the corresponding elements in each of our matrices in the left-hand side, is gonna be equal to zero. And that’s because on the right-hand sides the corresponding matrix is zero, as with the same for each of the other elements of each of our matrices.
So, as we said, our first equation is gonna be negative four 𝑥 minus seven 𝑦 plus 12 equals zero. Then, for our second equation, we’ve got six 𝑥 plus 𝑙𝑦 minus four equals zero. Then, we have 10𝑥 plus zero 𝑦 plus 40 equals zero. Well, we can disregard the zero 𝑦. So, what we’re left with is 10𝑥 plus 40 equals zero. Then, finally, for our fourth equation, we’ve got 𝑘𝑥 minus five 𝑦 minus eight equals zero. Okay, great. So, we’ve got our four equations.
So now, what we want to do is solve some of them to find 𝑥, 𝑦, 𝑘, and 𝑙. Well, the first one that looks like it’s gonna be the easiest to solve, and that’s because it’s only got one variable, is equation three. Well, if we’ve got 10𝑥 plus 40 equals zero and subtract 40 from each side of the equation, we get 10𝑥 equals negative 40. And then divide through by 10, we get 𝑥 is equal to negative four. Okay, great. So, we’ve found our 𝑥-value.
Well, now, what we’re gonna do is substitute our value for 𝑥 into equation one. So, when we do that, what we’re gonna get is negative four multiplied by negative four minus seven 𝑦 plus 12 equals zero. So, this is gonna give us negative seven 𝑦 plus 28 equals zero. So then, if we add seven 𝑦 to each side of the equation, we get 28 equals seven 𝑦. And then divide through by seven, you get four equals 𝑦. So, we’ve now found out what 𝑦 is.
So now, what we want to do is to find 𝑘. And to find this, what we’re gonna do is substitute 𝑥 equals negative four and 𝑦 equals four into equation four. And when we do that, we get negative four 𝑘 minus five multiplied by four minus eight equals zero, which gives us negative four 𝑘 minus 28 equals zero. So then, we can add 28 to each side. So, we get negative four 𝑘 equals 28. So then, if we divide through by negative four, we get 𝑘 equals negative seven. Okay, great. Now, all we need to do is find 𝑙.
Well, to find 𝑙, what we need to do is substitute 𝑥, 𝑦, and 𝑘 into equation two. So, when we do this, we’re gonna get six multiplied by negative four plus four 𝑙 minus four equals zero, which would give us negative 28 plus four 𝑙 equals zero. Then, we can add 28 to each side of the equation. And we get four 𝑙 equals 28. Divide through by four, and we get a value of 𝑙 of seven. So therefore, what we can say is the values of 𝑥, 𝑦, 𝑘, and 𝑙 that satisfy our given equation are negative four, four, negative seven, and seven, respectively.
