Fundamentals of Differential Equations (9th Edition)
9th Edition
ISBN: 9780321977069
Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider
Publisher: PEARSON
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5. Given the matrix
A=[²1]
(a) Find the value of x that makes A singular.
(b) Find the value of x that makes A-1 = A (i.e, that makes the matrix equal to its own
inverse).
15. If A=|
B =
0 1
and C =
1 0
, then find the matrix (A v B) 0 c
Properties of the Transpose In Problems 39–42, either prove
the properties in general using the fact that [ a, ]T = [a, ).
or demonstrate the properties for general 3 × 3 matrices.
39. (A")T = A
40. (A + B)T = AT + BT
41. (kA) = kAT, for any scalar k
42. (AB)" = BTAT
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- Problem 0.6 Given a symmetric matrix A for A = [8], QA(x, y) = [x_y] [0]. (1) Check by a direct computation that QA(x, y) = ax² + 2bxy + cy². (2) Explain what {(x, y) = R² : Q₁(x, y) = 1} represents geometrically, [1], and [2], respectively. 0 (3) Diagonalize the matrix define 3 B=11 3 Can you find an orthogonal matrix V such that VBV-¹ becomes diagonal? (4) Sketch the set {(x, y) = R² : 3x² + 2xy + 3y² = 1}, by relating it with QB, and using the ingredients from (1)-(3).arrow_forward1 Suppose A is a 4x4 non-singular matrix where det (A) = 3 What is the value of det (A-¹) ?arrow_forward2. factor 1/4, then reflects about the line y = x. 01 1/4 0 0 1/4 (a) A = (d) A = (₁ Find the standard matrix for the operator on R² which contracts with ( :) :) 1/4 0 0 1/4 (1¹4) (e) None of these (b) A = 0 1 -(;!) 0 (c) A =arrow_forward
- 3. Solve the following systems and find a fundamental matrix for each system: x-(; * x -{; (a) X' = (b) хarrow_forward3. (a) Solve the matrix equation for X: -1 0 1 1 2 1 1 -3 1 -1 | 3 1 1 1 3 (b) If possible, express A = 2 3 as a product of elementary matrices..arrow_forwardIn Problems 5–9, find a fundamental matrix for the system x' (t) = Ax(t) for the given matrix A. 5 -1 5. А %3D 8. -2 -2 6. А %3D -1 -1 -4 | |arrow_forward
- 4. Suppose that A and B are invertible matrices and c is a scalar. Solve the following equation for the matrix X, and simplify your answer as much as (but not more than!) possible: (b) AX+CX= A + B. 3 0 2 (*+2 3+3) - (39) 7 T where T indicates Determine x, y, z such that 2 the transpose of the matrix.arrow_forward20 1. solve the matrix equation AXAT adj(A) for XER. Justify your answer !arrow_forward3. 1 Given the matrix D = 2 3 1 -3 -2 4 -2 " find the inverse matrix of D.arrow_forward
- (2 1) Problem 8.1.35: Suppose Ā = ( 6 3). Verify that the matrix B = AA" is 2 5. symmetric.arrow_forward3. Diagonalize the matrix, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A = PDP^−1. You do not need to show your work for any row reduction but do you need to show all other work.arrow_forward13. Find all value of h such that the augmented matrix (h+1) (h+2) 0 0 (h-2) (h-2) 0 0 (h² - 1) represents a system with: (a) no solutions; (b) a unique solution; (c) infinitely many solutions.arrow_forward
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