Question Video: Using the Properties of Determinants to Evaluate the Determinant of a 3 × 3 Matrix | Nagwa Question Video: Using the Properties of Determinants to Evaluate the Determinant of a 3 × 3 Matrix | Nagwa

# Question Video: Using the Properties of Determinants to Evaluate the Determinant of a 3 × 3 Matrix Mathematics • First Year of Secondary School

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Find the value of the determinant of [4, 1, −8 and −6, 3, 6 and 0, 0, 0].

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### Video Transcript

Find the value of the determinant of the three-by-three matrix four, one, negative eight, negative six, three, six, zero, zero, zero.

In this question, we’re asked to evaluate the determinant of a given three-by-three matrix. We can do this by expanding over any of the rows or columns. And this would work. We can use this to get the correct answer. However, we can notice something interesting about this matrix. We can see that the third row contains only zeros.

Therefore, if we were to expand over the third row, every term in our expansion would have a factor of zero, which means the determinant would evaluate to give us zero. In fact, this result holds true in general. If we have a square matrix where any of the rows or columns only contains zero, then we can conclude the determinant of that matrix must be equal to zero. And in particular, we were able to show the determinant of the three-by-three matrix four, one, negative eight, negative six, three, six, zero, zero, zero is equal to zero because it has a row only containing zeros.

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