Question Video: Finding the Unknown Elements of a Singular Matrix Mathematics

Video Transcript

Find the value of 𝑥 that makes the matrix negative one, three, negative three, negative 𝑥, negative three 𝑥, 𝑥 plus one, negative five, negative five, negative five singular.

Well, the first thing we need to remind ourselves is what is a singular matrix? Well, a singular matrix is a matrix that doesn’t have an inverse. And what we also know is that a matrix is said to be singular if the determinant of the matrix is equal to zero. So the first thing we need to remind ourselves is how we find the determinant of a three-by-three matrix. So if we got the matrix 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, then the determinant of this matrix is equal to 𝑎 multiplied by the minor 𝑒, 𝑓, ℎ, 𝑖 — or the determinant of the two-by-two submatrix 𝑒, 𝑓, ℎ, 𝑖 — minus 𝑏 multiplied by the determinant of the two-by-two submatrix 𝑑, 𝑓, 𝑔, 𝑖 plus 𝑐 multiplied by the determinant of the two-by-two submatrix 𝑑, 𝑒, 𝑔, ℎ.

And just to remind us how we found our minors or two-by-two submatrices, then the way we find these is if we take the value that we’re looking at, so the element, so in this case, 𝑎 is the first element in the first row in the first column, then we delete the elements in that row and column. Then what we’re left with is the two-by-two matrix 𝑒, 𝑓, ℎ, 𝑖. Okay, great! So now we know what we need to do to find the determinant. Let’s find the determinant of our matrix.

So the determinant of our matrix is going to be equal to negative one multiplied by the determinant of the two-by-two submatrix negative three 𝑥, 𝑥 plus one, negative five, negative five, remembering that we found the submatrix by looking at the negative one, our first element in the first row and column, then deleting that row and column, and then we’ve got the four elements left over are the four elements in our two-by-two submatrix. Then what we have is minus three multiplied by the determinant of the two-by-two submatrix negative 𝑥, 𝑥 plus one, negative five, negative five. Then finally, we subtract three multiplied by the two-by-two submatrix, the determinant for, negative 𝑥, negative three 𝑥, negative five, negative five.

So now, the next thing we need to do is remind ourselves how we find the determinant of a two-by-two matrix. Well, if we got the matrix 𝑎, 𝑏, 𝑐, 𝑑, then the determinant is equal to 𝑎𝑑 minus 𝑏𝑐. So what we do is we cross multiply and then we subtract. So then what we’re gonna get is negative one multiplied by and then we’ve got negative three 𝑥 multiplied by negative five, which gives us 15𝑥, then minus and we’ve got negative five multiplied by 𝑥 plus one. Then we’re gonna subtract three multiplied by then we’ve got negative 𝑥 multiplied by negative five, which gives us five 𝑥, minus negative five multiplied by 𝑥 plus one. Then finally, we subtract three multiplied by five 𝑥 minus 15𝑥.

So then, if we distribute across our parentheses and simplify, we’re gonna get negative one multiplied by 15𝑥 plus five 𝑥 plus five minus three multiplied by five 𝑥 plus five 𝑥 plus five minus three multiplied by negative 10𝑥, which is gonna give us negative 20𝑥 minus five minus 30𝑥 minus 15 plus 30𝑥. Well, the negative 30𝑥 and the positive 30𝑥 cancel each other out. So then we’re left with negative 20𝑥 minus 20. And this is the determinant of our matrix. Okay, great!

Well, what we want to do now is set this equal to zero because what we want to do is find the value of 𝑥 that makes our matrix singular. So what we’ve got is zero equals negative 20𝑥 minus 20. So what we’ve got is negative 20𝑥 minus 20 equals zero. So then, if we divide through by negative one, we get 20𝑥 plus 20 equals zero. So then if we divide through by 20, what we’d get is 𝑥 plus one equals zero. So therefore, 𝑥 is equal to negative one. So what we can say is that the value of 𝑥 that makes matrix negative one, three, negative three, negative 𝑥, negative three 𝑥, 𝑥 plus one, negative five, negative five, negative five singular is negative one.