### Video Transcript

Is the matrix three, one, negative three, negative one invertible?

So, the first thing we need to look at is, what does invertible mean? Well, a matrix is said to be invertible or non-singular, which is its other name, if it has an inverse. Well, what we can do is we can use the determinant to work out whether a matrix is, in fact, invertible.

So, letβs consider if we had the matrix π΄ which is π, π, π, π. Well then, we have a general form for the inverse of that matrix, which is one over ππ minus ππ multiplied by the matrix π, negative π, negative π, π. If we have the condition that ππ minus ππ is not equal to zero. And thatβs because if ππ minus ππ was equal to zero, then this would mean that one over ππ minus ππ would be undefined. So, how is this gonna help us when I mention the determinant?

Well, we know that the determinant of matrix π΄ is equal to the determinant of π, π, π, π, which is equal to ππ minus ππ. Well, if we take a look back, we can see that this is the same as the ππ minus ππ, which is under the one. And weβre also told that for the inverse, ππ minus ππ, cannot be equal to zero. So therefore, we can surmise that a matrix must be invertible if the determinant of π΄ is not equal to zero. Okay, great! So, now we know what to do. Letβs work out whether the matrix is, in fact, invertible.

So, the first thing we need to do is work out the determinant of the matrix three, one, negative three, negative one. So, first of all, what weβre gonna do is multiply our π by our π, so itβs the top-left term by the bottom-right term, so three multiplied by negative one. And then, subtract one multiplied by negative three. Thatβs the top-right term multiplied by the bottom-left term. So, what weβre gonna get is negative three minus negative three. Well, if you subtract a negative, then it turns positive, so you got negative three add three. So then, we get a result of zero.

So, this is gonna help us to determine whether it is invertible. We can say that the matrix three, one, negative three, negative one is not invertible. And that is because it is singular. So therefore, it does not have an inverse. And we know that because the determinant of our matrix is equal to zero.