Video Transcript
By considering the value of the determinant, determine whether the matrix one, two, three, zero, two, one, three, one, zero has an inverse. If so, find the inverse by considering the matrix of cofactors.
A matrix has an inverse if, and only if, the value of its determinant is not equal to zero. So, to help us decide whether this matrix has an inverse, we begin by calculating the value of its determinant. Finding the determinant of a three-by-three matrix is a little more complicated than finding the determinant for a two-by-two matrix. So, we’ll need to be really careful carrying out each step.
The formula for the determinant of a matrix 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ, 𝑖 is 𝑎 multiplied by the determinant of the matrix 𝑒, 𝑓, ℎ, 𝑖 minus 𝑏 multiplied by the determinant of the matrix 𝑑, 𝑓, 𝑔, 𝑖 plus 𝑐 multiplied by the determinant of 𝑑, 𝑒, 𝑔, ℎ. Essentially, we take each of the elements in the top row and we multiply them by their minor. This is the determinant of the two-by-two matrix that’s left when we cover up the row and column that our element lies in.
So, for example, with the element 𝑎, we can see its minor is the determinant of the matrix 𝑒, 𝑓, ℎ, 𝑖. We also have to remember to multiply the second element 𝑏 by negative one when doing this. For our matrix, it’s one multiplied by the determinant of two, one, one, zero. And that’s because when we cover up the row and column that this first element one is in, we’re left with those four digits, two, one, one, zero.
We then subtract two multiplied by the determinant of the remaining two-by-two matrix zero, one, three, zero. And we add three multiplied by the determinant of zero, two, three, one. Next, we need to find the determinants of the two-by-two matrices. To do this, we find the product of the elements on the top left and bottom right. And we subtract the product of the elements on the top right and bottom left.
For this matrix, that’s two multiplied by zero minus one multiplied by one, which is negative one. For our second matrix, it’s zero multiplied by zero minus one multiplied by three, which is negative three. And for our third matrix, that’s zero multiplied by one minus two multiplied by three, which is negative six. This gives us a value of negative 13. Since the value of the determinant is not equal to zero, it’s negative 13. It does indeed have an inverse.
Now that we know that, we can find the inverse. And we need to do that by considering the matrix of cofactors. To do that, we begin by replacing each entry, or each element, in the matrix by its minor. For the first element, that’s the determinant of the two-by-two matrix two, one, one, zero. That’s negative one. And for the second element, that’s the determinant of the two-by-two matrix zero, one, three, zero. We’ve crossed out the middle column. That’s negative three. For the third element, it’s the determinant of the matrix zero, two, three, one, which is negative six.
We continue this process. We cover up the row and column that the entry is in. And we find the determinant of the two-by-two matrix that remains. So, our matrix is negative one, negative three, negative six, negative three, negative nine, negative five, negative four, one, two.
We create the matrix cofactors by changing the signs according to this pattern. We multiply the second element in the first row by negative one. And in fact, we do that for every other element. We do it for the first element in the second row, the third element in the second row, and the second element in the third row. So, our matrix of cofactors is negative one, three, negative six, three, negative nine, five, negative four, negative one, two.
Our third step is to transpose this matrix. Essentially, we reflect each of the elements in this diagonal. And we can see that the elements on the diagonal themselves must remain the same. We’ll swap these two threes. We’ll replace negative four with negative six and vice-a-versa. And we’ll switch five and negative one.
Our final step is to multiply by one over the determinant. Now you might be able to see why it was important that the determinant of our matrix was not equal to zero. If it had been, we would have been calculating one divided by zero, which we know is undefined. Another way of thinking about this is that we divide each element in the matrix by the determinant, by negative 13.
Negative one divided by negative 13 is one thirteenth. Three divided by negative 13 is negative three thirteenths. Negative four divided by negative 13 is four thirteenths. And we’ll repeat this process to get the inverse shown.
