# Video: Evaluating Determinants Using Their Properties

Find the value of [β3π₯ + 3, 4π₯Β² + 4 and β3π¦ + 4, βπ¦Β² + 3].

02:54

### Video Transcript

Find the value of the determinant of the matrix given by negative three π₯ plus three, four π₯ squared plus four, negative three π¦ plus four, negative π¦ squared plus three.

Remember, for a two-by-two matrix π΄ with elements π, π, π, π, the determinant of this matrix can be found by subtracting the product of elements π and π from the products of elements π and π.

In the matrix in our question, π is negative three π₯ plus three. π is four π₯ squared plus four. π is negative three π¦ plus four. And π is negative π¦ squared plus three. Weβre then going to find the product of elements π and π. Thatβs negative three π₯ plus three multiplied by negative π¦ squared plus three. And from this, weβre going to subtract the product of elements π and π. Thatβs four π₯ squared plus four multiplied by negative three π¦ plus four.

We can use our preferred method to expand each of these brackets. Iβm going to use the FOIL method. Remember βFβ stands for first. We multiply the first term in the first bracket by the first term in the second. Negative three π¦ multiplied by negative π¦ squared is three π₯π¦ squared. We then multiply the outer terms. Negative three π₯ multiplied by three is negative nine π₯.

βIβ stands for inner. So weβre going to multiply the inner terms. And three multiplied by negative π¦ squared is negative three π¦ squared. And finally, weβre going to multiply the last term in each bracket. Three multiplied by three is nine.

Letβ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 12π₯ squared π¦. Four π₯ squared multiplied by four is 16π₯ squared. Multiplying the inner two terms gives us four multiplied by negative three π¦, which is negative12π¦. And finally, four multiplied by four is 16.

Weβre going to simplify this expression by collecting like terms where we can, noting that weβre going to be subtracting everything in this second bracket. Our first three terms are three π₯π¦ squared minus nine π₯ minus three π¦ squared. Then, nine minus 16 gives us negative seven. Weβre subtracting negative12π₯ squared π¦. So it gives us plus 12π₯ squared π¦. Weβre subtracting 16π₯ squared. And then, we subtract negative 12π¦. So thatβs the same as adding 12π¦. Itβ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 π₯π¦ squared minus nine π₯ minus three π¦ squared plus 12π₯ squared π¦ minus 16π₯ squared plus 12π¦ minus seven.