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 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

41. Remember, det(A) = Σ (sgn P)(prod P). In this matrix, two nonzero patterns exist: (2 -> 3 -> 1 -> 2 -> 4), with 5 inversions, and (2 -> 3 -> 3 -> 2 -> 2), with 8 inversions.

       det(A) = (-1)5(2 * 3 * 1 * 2 * 4) + (-1)8(2 * 3 * 3 * 2 * 2)

                   = (-48) + (72) = 24


6.3

22. Cramer's Rule states that in the system Ax = b , where A is an invertible n x n matrix, the components xi of the solution vector are xi = det(Ab,i)/det(A), where Ab,i is the matrix obtained by replacing the ith column of A with b

x1 = det([[1  7][3 11]])/det(A) = (-10) / (5) = -2

x2 = det([[3  1][4  3]])/det(A) = (5) / (5) = 1

x = [-2  1]

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