Video Transcript
Given that the matrix negative three, negative two, one, one multiplied by the matrix negative one, negative two, one, 𝑥 is equal to 𝐼, where 𝐼 is the unit matrix, determine the value of 𝑥.
Well, the first thing we need to do is remember what the unit matrix is, which is denoted here by 𝐼. Well, the unit or identity matrix is a matrix whose diagonals from the top left to the bottom right are all one and then any other elements are all zero. So, in this question, because we’re dealing with two-by-two matrices, our unit matrix is going to be the matrix one, zero, zero, one.
So, the next thing we need to do is remind ourselves how we’d multiply two matrices together. So, imagine we’ve got the two matrices 𝑎, 𝑏, 𝑐, 𝑑 and 𝑒, 𝑓, 𝑔, ℎ. Well, then what we do for the first element is multiply first of all the first element in the first row of the first matrix by the corresponding first element of the first column in the second matrix, so we get 𝑎𝑒. And then, we add to this the second element in the first row of the first matrix multiplied by the second element in the first column of the second matrix, so we get 𝑏𝑔.
So now, what we do is we move along. So, we have 𝑎𝑓 plus 𝑏ℎ. And we get this because we have the first element in the first row of the first matrix multiplied by the first element in the second column of the second matrix. And then, we go for the second element in the first row of the first matrix multiplied by the second element in the second column of the second matrix. So, for the bottom row, what we do is carry on the same pattern, and we get 𝑐𝑒 plus 𝑑𝑔 and then 𝑐𝑓 plus 𝑑ℎ.
Okay, great. So, we’ve now reminded ourselves how we multiply two matrices together. So, let’s get on and do that. So, what we’re gonna get for our first element is negative three multiplied by negative one plus negative two multiplied by one. Then, for the next element, it’s gonna be negative three multiplied by negative two plus negative two multiplied by 𝑥. Then, if we move to the next row, we’ve got one multiplied by negative one plus one multiplied by one. And then, for the final element, we’ve got one multiplied by negative two plus one multiplied by 𝑥.
Okay, great. So now, all we need to do is work these out. So, our first element is gonna be one cause we have negative three multiplied by negative one, which is three. And then, we’ve got add negative two. So, it’s just gonna be one. Then, our next element’s gonna be six minus two 𝑥, then zero because you got one multiplied by negative one, which is negative one, add one, which is just zero. And then, finally, our final element is negative two plus 𝑥. Okay, great. But what’s our next step?
Well, what we were told is that everything on the left-hand side, so the calculation on the left-hand side, so the multiplication of the two matrices, is equal to our unit or identity matrix. So, we can say that the matrix one, six negative two 𝑥, zero, negative two plus 𝑥 is equal to the matrix one, zero, zero, one. So therefore, we can use either of the elements that have 𝑥 in to find out what 𝑥 is gonna be; then we can check it in the other element.
So, let’s start with the top-right element of each of our matrices because what we can do is equate these values. So, six minus two 𝑥 is equal to zero. So, if we got six minus two 𝑥 is equal to zero, if we add two 𝑥 to each side, we get six equals two 𝑥. And then, divide through by two, we get three is equal to 𝑥. So then, what we’ve done is found our value of 𝑥 to be three. So, we can say 𝑥 is equal to three.
And what we can do is double-check this by looking at the next element below. Well, if we equate the bottom right-hand side elements from each of our matrices, we’re gonna get negative two plus 𝑥 equals one. So then, if we add two to each side of the equation, what we’re gonna get is 𝑥 equal to three. And this is what we got when we looked at the other element. So therefore, we can definitely say that the value of 𝑥 is three.
