Question Video: Using the Properties of Determinants to Find One Determinant from Another Mathematics

Consider the equation |6, −20, 11 and −14, −18, 14 and −16, 1, 2| = 298. Find, without expanding, the determinant |2, −16, 1 and 14, −14, −18 and 11, 6, −20|.

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Video Transcript

Consider the equation the determinant of the matrix with elements six, negative 20, 11; negative 14, negative 18, 14; and negative 16, one, two is equal to 298. Find, without expanding, the determinant of the matrix with elements two, negative 16, one, 14, negative 14, negative 18, 11, six, negative 20.

In order to find the given determinant, we can use the property of determinants, telling us that if any two rows or columns of an 𝑛-by-𝑛 matrix are interchanged, then the determinant of the matrix changes sign. Considering the given equation, if we call the matrix 𝐴, then the determinant of matrix 𝐴 is 298.

Now, if we call the matrix we want to find the determinant of 𝐵, we see that, although in different places, the elements in rows one and row three of matrix 𝐵 are the same as those in row three and row one of matrix 𝐴. And so our first step is to interchange row three and row one of matrix 𝐴. And by our property of determinants, since we’ve interchanged rows, we must change the sign of our determinant. Our determinant is now negative 298.

Next, if we interchange column two and column three, we must again change the sign of our determinant. So again, our determinant is 298. Now, finally, if we interchange column one and column two, we must again change the sign of our determinant. So it’s now negative 298.

So, by interchanging rows and columns, we’ve reached the determinant of the matrix 𝐵. The determinant of the given matrix is therefore negative 298. And it’s worth noting that we could’ve done these operations in any order to reach the same answer.

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