Video: Evaluating Determinants Using Their Properties

Find the value of [βˆ’3π‘₯ + 3, 4π‘₯Β² + 4 and βˆ’3𝑦 + 4, βˆ’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 𝑦 plus four, negative 𝑦 squared plus three.

Remember, for a two-by-two matrix 𝐴 with elements π‘Ž, 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 π‘Ž and 𝑑.

In the matrix in our question, π‘Ž is negative three π‘₯ plus three. 𝑏 is four π‘₯ 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 π‘Ž and 𝑑. That’s negative three π‘₯ 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 π‘₯ 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 β€œF” 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 π‘₯𝑦 squared. We then multiply the outer terms. Negative three π‘₯ multiplied by three is negative nine π‘₯.

β€œI” 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 12π‘₯ squared 𝑦. Four π‘₯ squared multiplied by four is 16π‘₯ 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 π‘₯𝑦 squared minus nine π‘₯ minus three 𝑦 squared. Then, nine minus 16 gives us negative seven. We’re subtracting negative12π‘₯ squared 𝑦. So it gives us plus 12π‘₯ squared 𝑦. We’re subtracting 16π‘₯ 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 π‘₯𝑦 squared minus nine π‘₯ minus three 𝑦 squared plus 12π‘₯ squared 𝑦 minus 16π‘₯ squared plus 12𝑦 minus seven.

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