Chapter 6: Determinants
I will show several problems where I find the determinant, illustrating the several methods of doing this.
6.1
2. For any 2x2 matrix A, det(A) = ad - bc, so det(A) = (2)(5) - (3)(4) = -2 . Since this is not 0, A is invertible.
5. Let's use Laplace Expansion and expand across the first column. Remember to alternate signs.
det(A) = (1)(2)(det(A11) + (-1)(5)(det(A12) + (1)(7)(det(A13)
= (2 * 55) + (-5 * 0) + (7 * 0) = 110 The matrix is invertible
6. Determinant of a upper- or lower-triangular matrix is simply the product of the diagonal entries.
det(A) = (6)(4)(1) = 24 The matrix is invertible
8. For any 3x3 matrix A with column vectors u, v, w, determinant of A is u · vxw
det(A) = [1 1 3] · ([2 1 2] x [3 1 1])
= [1 1 3] · [-1 4 -1]
= 0 The matrix is not invertible.
