Use the properties of determinants to evaluate the determinant of the matrix with elements six, two, six, two.
In this question, we want to find the determinant of the two-by-two matrix with elements six, two in the first row and six, two in the second row. Now we could calculate this using the formula for the determinant of a two-by-two matrix with elements 𝑎, 𝑏, 𝑐, 𝑑; that is, 𝑎𝑑 minus 𝑏𝑐. But we’re actually asked to use the properties of determinants, not the formula.
If we consider our matrix, we can see that the rows have the same elements — that’s six and two — and they’re in the same order. So, since our rows 𝑅 two and 𝑅 one are the same, we can apply the property of determinants that tells us if a matrix has a repeated row or column, then the determinant of the matrix is equal to zero. Applying this to our matrix then, we see that the determinant of the matrix with elements six, two, six, two is zero.
Of course, we can check this by using the formula. With 𝑎 is equal to six, 𝑏 is two, 𝑐 is six, and 𝑑 is two, 𝑎𝑑 is six multiplied by two, and 𝑏𝑐 is two multiplied by six. Our determinant is 12 minus 12, which is equal to zero as expected.