Question

Differention Equations - Can someone answer the checked numbers please?

Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve for A1, noting that ao- det A0 The matrix A-0 22 has characteristic equation 0 0 2 2-A)P-8-12A +62- 0, so 8A1-12+6A -A, r 8A1-12 Hence we need only divide by 8 after computing 6A+. 23 1 4 12 10 4 -64 EXERCISES 1. Find AB, BA, and the determinants of A, B, AB, and BA when (a) -2 2 -3 2 0 0 -1 0 2. For the matrices in Exercise 1b, what are det (2A) and det (28)7 3. A diagonal matrix D is a square matrix such that only the main diagonal entries d-du are allowed to be nonzero. We sometimes write D diag d). Show that det D - dd2 In particular, det -1. v 4. What is the relation between det A and det (-A)? 5. Verify the product rule for the pairs of matrices (a) and b) in Exercise 1. 6. Apply the product rule to show that if A is invertible, then det A 0 and + n) by which has A in the upper left comer, B (det A)-1 det A) 7. Let A be an m by m matrix and B an n by n matrix. Consider the ( A 0 )matrix ( ), (m+ n in the lower right corner, and zeros esewhere. Show that its determinant is equal to (det AXdet B. uint. Consider the cases A-1 and B = L Then use the product rule.] 660 APTENDIX Matrix Aigebra &. Use the method of Example 17 of the text to evaluate 0 1 2 -1 (b) de 1 3 9 27 o) det 1 20 1 4 16 2 -1 0 9. (a) Compute 0 -15 6 0 0 31 (b) A matrix A, like the one in part (a), in which every element below the diagonal is 0, is said to be upper triangular. Show that if A is any trian- gular matrix, then det A is equal to the product of the diagonal elements 10. Let A be the 3 by 3 matrix -1 3 7 (a) what is A? (b) Compate det A and det A 11. Show that for an n by n matrix A det (AR') - det (AA) (det A)" 12. (a) Show that if A, B, and C are n by n matrices such that then B- C. [Hint: Multiply CA -I by B on the right. (b) Show that if A and B are n by n matrices such that AB-I, then A and B are both invertible, with A B and BA. Hint: Use the product rule for deterninants, together with part (a)] 13. Use Theorem B.9 to determine which of the following matrices have in- verses and then use either Theorem B.9 or the remark following the Cayley- Hamilton theorem to find the inverses of the ones that are invertible. (a) 3 15 凵(d)| 0 01, treal. 2 -2 0 1 (b)-1 1 0 03 3 0 0 3 1 -1 (2-4-7) (e)1 0 0

Answer 1

3 2.-3 3 1 3+ 2 3-3 !:牝"ЂЕ 기.L: :/ 3 2 3 31 BAI =-3x7-(-74) =