Question Video: Finding the Cofactor of a Matrix Mathematics • Third Year of Secondary School

Video Transcript

Find the cofactor matrix of matrix 𝐴 equals seven, negative five, negative eight, negative three, negative seven, negative two, zero, negative four, negative eight.

So, in a problem like this, where we want to find the cofactor of a matrix, then what we want to do first of all is find the minors of each of our elements in this matrix. Well, first, we’ll just remind ourselves what we need to do if we want to find the minor of an element. Well, I’ve just drawn another matrix here 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ, 𝑖. Well, now, if we want to find the minor of element 𝑎, then what we do is we delete the column and row that element 𝑎 is in. And then our minor is the determinant of the four elements that are left behind. So it’s gonna be determinant of the two-by-two matrix 𝑒, 𝑓, ℎ, 𝑖. Okay, great. So we’ve reminded ourselves what a minor is. So now let’s find the minors of our elements in our matrix.

So now if we have a matrix made up with the minors of matrix 𝐴, I’ve called it 𝐴 sub 𝑚, this is equal to — then we’ve got the determinant of negative seven, negative two, negative four, negative eight. Then we’ve got negative three, negative two, zero, negative eight; negative three, negative seven, zero, negative four; negative five, negative eight, negative four, negative eight; seven, negative eight, zero, negative eight; seven, negative five, zero, negative four; negative five, negative eight, negative seven, negative two; seven, negative eight, negative three, negative two; and finally the determinant of seven, negative five, negative three, negative seven.

So now, before we work out the values of our determinants, what we need to do is remember the sign rule. And the sign rule which helps us with this tells us that we have elements, and this is the signs that they need to take, positive, negative, positive; negative, positive, negative; and then positive, negative, positive. So what we can do is add these in front of the different elements we’ve got in our matrix of our minors. But if we put these signs in front of our different elements in our matrix, then what we actually do is now have the cofactor of our matrix 𝐴. So all we need to do now is work out the values for each of the determinants of our minors.

So now, just to remind us how we work out the values for the determinants of our minors, then if we take a look at the right-hand side, we’ve got the minor 𝑒, 𝑓, ℎ, 𝑖. Then this is equal to 𝑒𝑖, so 𝑒 multiplied by 𝑖, minus 𝑓 multiplied by ℎ. So to give us an example of this, we’ll take a look at our first element. So what we’d have is negative seven multiplied by negative eight minus negative two multiplied by negative four, which would give us 56 minus eight, which would be 48. And it would stay positive because we can see that following our sign rule, the first element is going to be positive. Then we’ll just do the next one as another example before filling them all in.

So for our next element, what we’re gonna have is negative three multiplied by negative eight minus negative two multiplied by zero, which gonna give us 24 minus zero, which is equal to 24. So is that our next element? Well no, and this is where we need to make sure that we don’t make any mistakes in our calculations because if we take a look at the sign rule, our second element should be negative. So therefore, the value we want is negative 24. So then we can use this method for the rest of our elements. And when we do, what we’re gonna have is the cofactor matrix of matrix 𝐴, which is equal to seven, negative five, negative eight, negative three, negative seven, negative two, zero, negative four, negative eight. And this cofactor matrix is gonna be the matrix 48, negative 24, 12, negative eight, negative 56, 28, negative 46, 38, and negative 64.