### Video Transcript

Find the set of real values of 𝑥

that make the two-by-two matrix 𝑥 minus three, eight, two, 𝑥 plus three

singular.

Let’s begin by defining the word

singular when it comes to matrices. We say that a matrix is singular if

it’s not invertible; it doesn’t have an inverse. We know that a matrix is invertible

if its determinant is not equal to zero, and the converse is also true. So, in other words, a matrix is

singular if its determinant is equal to zero. In this case then, we need to find

the set of real values of 𝑥 such that the determinant of our matrix is equal to

zero. The determinant of a two-by-two

matrix 𝑎, 𝑏, 𝑐, 𝑑 is 𝑎𝑑 minus 𝑏𝑐. We subtract the product of the

elements in the top right and bottom left from the product of those in the top left

and bottom right.

So, in this case, that’s 𝑥 minus

three times 𝑥 plus three minus eight times two. If we distribute these parentheses,

we get 𝑥 times 𝑥, which is 𝑥 squared, plus three 𝑥 minus three 𝑥 minus three

times three, which is nine. That simplifies to 𝑥 squared minus

nine. And eight multiplied by two is

16. So the determinant of our matrix is

𝑥 squared minus nine minus 16, which is 𝑥 squared minus 25. We’re trying to find the set of

values of 𝑥 that make our matrix singular. In other words, which values of 𝑥

make the determinant zero? So, let’s set our expression for

the determinant equal to zero and solve for 𝑥. That is, 𝑥 squared minus 25 equals

zero.

Adding 25 to both sides of this

equation gives us 𝑥 squared equals 25. And then we’ll take the square root

of both sides of our equations, remembering to take the positive and negative square

root of 25. That gives us 𝑥 is equal to

positive or negative five. We can use these squiggly brackets

to help us represent the set of values that make our matrix singular. They are negative five and

five. Note that at this stage, we could

check our solutions by substituting each value of 𝑥 into our original matrix and

then checking that the determinant is indeed equal to zero.