Video Transcript
Use the properties of determinants to evaluate the sum of the determinant of the matrix with elements negative four, five, 13, two, six, negative two, and five, 14, negative one with the determinant of the matrix with elements negative four, negative 13, 13, two, negative two, negative two, and five, negative four, negative one.
We’re given a sum involving the determinants of two three-by-three square matrices. And to evaluate this sum, we recall a property of determinants that tells us that if two determinants differ by just one column, we can add the determinants together by simply summing these two columns. The matrices shown have different first columns; however, their second and third columns are the same. And so the sum of their determinants is the determinant of the matrix whose first column is the sum of their two first columns and the remaining columns stay the same.
In the given expression, we note that the first column of our two determinants is the same, that is, with elements negative four, two, and five. And similarly, the third columns are the same with elements 13, negative two, and negative one. And so the two determinants differ only by their second columns. And so now applying the property of determinants we noted earlier, the first element in the second column of our determinant is five plus negative 13, the second element is six plus negative two, and the third element is 14 plus negative four. So now the sum of the determinants is the determinant of the matrix with elements negative four, negative eight, 13, two, four, negative two, and five, 10, negative one.
Now we can also use another property of determinants. That is, if two rows or columns of a determinant are proportional to each other, then the determinant is equal to zero. This means that if one row or column is a constant multiple of another, then the determinant is equal to zero. And in our new sum determinant, we see that column two is actually two times column one. We have that negative eight is two multiplied by negative four, four is equal to two multiplied by two, and 10 is equal to two times five. And so since column two is equal to two times column one, our determinant is equal to zero. Using the properties of determinants then, we found that the sum of the given determinants is zero.
