Question Video: Finding an Unknown by Evaluating the Determinant of a Triangular Matrix | Nagwa Question Video: Finding an Unknown by Evaluating the Determinant of a Triangular Matrix | Nagwa

# Question Video: Finding an Unknown by Evaluating the Determinant of a Triangular Matrix Mathematics • First Year of Secondary School

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Find the solution set of |𝑥, 0, 0 and −1, −5𝑥, 0 and 2, 1, 𝑥| = −80𝑥.

02:09

### Video Transcript

Find the solution set of the determinant of the matrix 𝑥, zero, zero, negative one, negative five 𝑥, zero, two, one, 𝑥 equals negative 80𝑥.

In order to find the solution set, we’ll need to begin by evaluating the determinant of this three-by-three matrix. And if we look carefully, we might notice that all of the elements above the leading diagonal in this matrix are zero. This means it’s a triangular matrix. And since it’s a triangular matrix, this will save us some time in calculating the determinant. Specifically, the determinant of a triangular matrix is the product of the entries on the leading diagonal.

In this case, the determinant of our matrix is the product of the elements 𝑥, negative five 𝑥, and 𝑥, which is negative five 𝑥 cubed. And so we can rewrite our original equation by replacing the determinant of the matrix with the expression negative five 𝑥 cubed. And we get negative five 𝑥 cubed equals negative 80𝑥. Remember, to solve this, we can’t just divide through by 𝑥 since we don’t know whether it’s equal to zero. Instead, we’ll add 80𝑥 to both sides and then factor the resulting expression. Factoring the left-hand side, and we get negative five 𝑥 times 𝑥 squared minus 16, and that’s equal to zero. And in fact, we can further factor the expression 𝑥 squared minus 16 using the difference of two squares. That gives us 𝑥 minus four times 𝑥 plus four.

So, for the product of these three expressions to be zero, we know that at least one of the expressions themselves must be zero. That is, negative five 𝑥 equals zero or 𝑥 minus four equals zero or 𝑥 plus four equals zero. Then, solving each equation for 𝑥, and we find that there are three solutions to our original equation. They are 𝑥 equals zero, 𝑥 equals four, and 𝑥 equals negative four.

Since we were asked to find the solution set, we’re going to write this in set notation. It’s the set containing the elements zero, four, and negative four.

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