University:
Whitman College
Course:
MATH-126 | Calculus II
Academic year:

2024
Views:

199

Pages:

1
Author:

Selah Carney

HW Solutions, 2.2 2.2, 16 Suppose that A, B are n × n, B and AB are invertible. Show that A is invertible. SOLUTION: Using the hint, if C = AB, then C is invertible, and CB −1 = ABB −1 = A Therefore, A is the product of the invertible matrix C and B −1 , so A is invertible. 2.2, 18 Suppose P is invertible and A = P BP −1 . Solve for B in terms of A. SOLUTION: Be sure you multiply on the correct side: A = P BP −1 ⇒ P −1 A = P −1 P BP −1 = BP −1 ⇒ P −1 AP = BP −1 P = B 2.2, 20 Suppose that A, B, X are n × n with A, X, and A − AX invertible. Suppose that (A − AX)−1 = X −1 B (a) Show that B is invertible: SOLUTION: Multiply on the left by X, and X(A − AX)−1 = B. Therefore, B is the product of invertible matrices (and is itself invertible). (b) Solve the equation given above for X. If you need to invert a matrix, explain why. SOLUTION: I don’t recall what we did in class (there are multiple ways of expressing the solution), but here is one possibility. Multiply both sides by (A − AX): X(A − AX)−1 = XX −1 B X = B(A − AX) = BA − BAX ⇒ ⇒ X(A − AX)−1 = B X + BAX = BA ⇒ ⇒ (I + BA)X = BA Is I + BA invertible? The last equality implies that I + BA = BAX −1 , which is the product of three invertible matrices. Therefore X = BA(I + BA)−1 Alternative Solution: Another solution might be the following- First, invert both sides of the given equation, since we know that B is invertible: A − AX = B −1 X A = B −1 X + AX = (B −1 + A)X ⇒ This last equation tells us that B −1 + A = AX −1 , so it is also invertible. Now do the inversion: X = (B −1 + A)−1 A Just for fun: Can you make the two solutions look alike? 1

HW Solutions, 2.2 - Assignment

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