Video Transcript
Use the properties of determinant to evaluate the determinant of the three-by-three matrix negative eight, negative nine, zero, six, two, zero, negative five, eight, zero.
In this question, we’re asked to evaluate the determinant of a given three-by-three matrix. And we’re told that we can do this by using the properties of determinants. This is because we could just evaluate the determinant of this three-by-three matrix by using the definition of a determinant. And this would give us the correct answer. However, we can simplify this process if we use one of the properties of determinants; in particular, we need to notice the third column of this matrix is all zeros. And we recall we can evaluate the determinant by expanding over any row or column of the matrix. And it’s usually a good idea to pick the row or column with the most zeros since this will make the most terms cancel.
In particular, we can recall the following definition of the determinants where we expand over the third column. The determinant of the three-by-three matrix 𝐴 is equal to the sum from 𝑖 equals one to three of negative one to the power of 𝑖 plus three multiplied by 𝑎 sub 𝑖 three times the matrix minor of 𝐴 at 𝑖 three. And in particular, we notice all of these terms share a factor of 𝑎 sub 𝑖 three, and these are the terms in the column we’re expanding over. In this case, this is all zeros. Therefore, if we applied this definition of the determinant to the matrix given in the question, we would see all of the terms have a factor of zero. So the determinant must be equal to zero.
And it’s worth pointing out this result holds true in general. If we have a square matrix where one of the rows or columns contains only zeros, then we can evaluate the determinant of this matrix by expanding over the zero row or column. Then every term in the determinant will have a factor of zero. So the determinant of this matrix is zero. Therefore, we were able to show the determinant of the matrix given to us in the question is equal to zero.