Video Transcript
Find the value of the determinant
of the matrix given by negative three 𝑥 plus three, four 𝑥 squared plus four,
negative three 𝑦 plus four, negative 𝑦 squared plus three.
Remember, for a two-by-two matrix
𝐴 with elements 𝑎, 𝑏, 𝑐, 𝑑, the determinant of this matrix can be found by
subtracting the product of elements 𝑏 and 𝑐 from the products of elements 𝑎 and
𝑑.
In the matrix in our question, 𝑎
is negative three 𝑥 plus three. 𝑏 is four 𝑥 squared plus
four. 𝑐 is negative three 𝑦 plus
four. And 𝑑 is negative 𝑦 squared plus
three. We’re then going to find the
product of elements 𝑎 and 𝑑. That’s negative three 𝑥 plus three
multiplied by negative 𝑦 squared plus three. And from this, we’re going to
subtract the product of elements 𝑏 and 𝑐. That’s four 𝑥 squared plus four
multiplied by negative three 𝑦 plus four.
We can use our preferred method to
expand each of these brackets. I’m going to use the FOIL
method. Remember “F” stands for first. We multiply the first term in the
first bracket by the first term in the second. Negative three 𝑦 multiplied by
negative 𝑦 squared is three 𝑥𝑦 squared. We then multiply the outer
terms. Negative three 𝑥 multiplied by
three is negative nine 𝑥.
“I” stands for inner. So we’re going to multiply the
inner terms. And three multiplied by negative 𝑦
squared is negative three 𝑦 squared. And finally, we’re going to
multiply the last term in each bracket. Three multiplied by three is
nine.
Let’s repeat this process for the
second set of brackets. Multiplying the first term from the
first bracket by the first term in the second bracket gives us negative 12𝑥 squared
𝑦. Four 𝑥 squared multiplied by four
is 16𝑥 squared. Multiplying the inner two terms
gives us four multiplied by negative three 𝑦, which is negative12𝑦. And finally, four multiplied by
four is 16.
We’re going to simplify this
expression by collecting like terms where we can, noting that we’re going to be
subtracting everything in this second bracket. Our first three terms are three
𝑥𝑦 squared minus nine 𝑥 minus three 𝑦 squared. Then, nine minus 16 gives us
negative seven. We’re subtracting negative12𝑥
squared 𝑦. So it gives us plus 12𝑥 squared
𝑦. We’re subtracting 16𝑥 squared. And then, we subtract negative
12𝑦. So that’s the same as adding
12𝑦. It’s not entirely necessary, but we
can move the constant to the end.
And then we see the determinant of
the matrix in our question is three 𝑥𝑦 squared minus nine 𝑥 minus three 𝑦
squared plus 12𝑥 squared 𝑦 minus 16𝑥 squared plus 12𝑦 minus seven.
