Answer Happy

4. Find the determinant of the following matrix by using the method shown in Example 1 in section 5.2. 3 -4 2 -2 7 3 -1 1

EXAMPLE 1 Find the eigenvalues of A = SOLUTION We must find all scalars λ such that the matrix equation Use this method (4-1)x= 0 ↑ = [3_-3] -6 has a nontrivial solution. By the Invertible Matrix Theorem in Section 2.3, this problem is equivalent to finding all such that the matrix A - I is not invertible, where L Recall that So 2 2-1 3 A 11 -^² = [3 -3]-[82] [²3^ _6²³x] -6-λ By Theorem 4 in Section 2.2, this matrix fails to be invertible precisely when its determinant is zero. So the eigenvalues of A are the solutions of the equation det(A - λI) = det = 3 x² [² =^_6²³² ₁] = 0 3 -6-λ det [a b] = d = ad-bc det(A − 1) = (2 − λ) (−6 − λ) – (3)(3) = −12+6λ- 2λ +1²-9 = λ² + 4λ - 21 = (λ− 3)(λ + 7) If det(A - λI) = 0, then λ = 3 or λ = -7. So the eigenvalues of A are 3 and -7.