Question Video: Finding the Inverse of a Matrix Using the Properties of Determinants Mathematics • Third Year of Secondary School

Video Transcript

Determine whether the matrix one, two, three, zero, two, one, two, six, seven has an inverse by finding whether the determinant is nonzero. If the determinant is nonzero, find the inverse using the inverse formula involving the cofactor matrix.

Well, the first thing we want to remind ourselves is how we would find the determinant of a three-by-three matrix. So, let’s consider the determinant of the three-by-three matrix 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ, 𝑖. Well, what we have is 𝑎 multiplied by then we’ve got a minor, or it’s also the determinant of a submatrix which is two by two, 𝑒, 𝑓, ℎ, 𝑖 minus 𝑏 multiplied by the determinant of the submatrix 𝑑, 𝑓, 𝑔, 𝑖 plus 𝑐 multiplied by the determinant of the submatrix 𝑑, 𝑒, 𝑔, ℎ.

We might think, well, where did you get the minors or submatrices from? Well, let’s have a quick look at the first one. Well, we took the element from the first row and first column, which is 𝑎. Well, then, if we delete the elements in its column and row, then what we’re gonna be left with is this two-by-two submatrix 𝑒, 𝑓, ℎ, 𝑖. And we used the same method for each stage of the process.

Okay, great, so we now know how to find the determinant of a three-by-three matrix. So, let’s do it, and let’s find the determinant of our matrix. So therefore, we’ve got the determinant of the matrix one, two, three, zero, two, one, two, six, seven. This is gonna be equal to one multiplied by the determinant of the two-by-two matrix two, one, six, seven minus two multiplied by the determinant of the two-by-two matrix zero, one, two, seven plus three multiplied by the determinant of the matrix zero, two, two, six.

Well, it’s at this point that we want to remind ourselves how we’d find the determinant of a two-by-two matrix. Well, if we’ve got the determinant of the matrix 𝑎, 𝑏, 𝑐, 𝑑, then what we do is cross multiply, so multiply 𝑎 by 𝑑 and 𝑏 by 𝑐, and then we subtract 𝑏𝑐 from 𝑎𝑑. Okay, great, so now let’s use this to work out the value of the determinant of our matrix.

So, what we’re gonna have is one multiply by then we’ve got 14 minus six minus two multiplied by zero minus two plus three multiplied by zero minus four, which is gonna be equal to eight plus four minus 12, which is gonna give us a value of zero. So therefore, we could say that the determinant is equal to zero.

So therefore, we can say in answer to the question, there is no inverse because the determinant is equal to zero.