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, let’s consider if we had the matrix 𝐴 which is 𝑎, 𝑏, 𝑐, 𝑑. Well then, we have a general form for the inverse of that matrix, which is one over 𝑎𝑑 minus 𝑏𝑐 multiplied by the matrix 𝑑, negative 𝑏, negative 𝑐, 𝑎. If we have the condition that 𝑎𝑑 minus 𝑏𝑐 is not equal to zero. And that’s because if 𝑎𝑑 minus 𝑏𝑐 was equal to zero, then this would mean that one over 𝑎𝑑 minus 𝑏𝑐 would be undefined. So, how is this gonna help us when I mention the determinant?
Well, we know that the determinant of matrix 𝐴 is equal to the determinant of 𝑎, 𝑏, 𝑐, 𝑑, which is equal to 𝑎𝑑 minus 𝑏𝑐. Well, if we take a look back, we can see that this is the same as the 𝑎𝑑 minus 𝑏𝑐, which is under the one. And we’re also told that for the inverse, 𝑎𝑑 minus 𝑏𝑐, cannot be equal to zero. So therefore, we can surmise that a matrix must be invertible if the determinant of 𝐴 is not equal to zero. Okay, great! So, now we know what to do. Let’s 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 we’re gonna do is multiply our 𝑎 by our 𝑑, so it’s the top-left term by the bottom-right term, so three multiplied by negative one. And then, subtract one multiplied by negative three. That’s the top-right term multiplied by the bottom-left term. So, what we’re 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.