# Question Video: Using the Properties of Determinants to Evaluate a 3 × 3 Determinant Mathematics

Use the properties of determinants to evaluate [−1, 0, 0 and −5, 5, 0 and 9, −4, −4].

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### Video Transcript

Use the properties of determinants to evaluate the determinant of the matrix with elements negative one, zero, zero, negative five, five, zero, and nine, negative four, negative four.

To evaluate the given determinant, we note that the matrix is a three-by-three lower triangular matrix. That is a three-by-three matrix whose elements above the main diagonal are all equal to zero. An upper triangular matrix, on the other hand, is a matrix whose elements are all zero below the main diagonal.

Now, one of the properties of determinants is that the determinant of either an upper or lower triangular matrix is the product of the diagonal elements. And applying this to the given determinant in the question, we see that the determinant is negative one multiplied by five multiplied by negative four, which is 20. Hence, the determinant of the matrix whose elements are negative one, zero, zero, negative five, five, zero, and nine, negative four, negative four is 20.