Question

Suppose A and B are invertible matrices. Mark each statement as true or false. True means that the statement is true for all invertible matrices A and B. a.(A + B)² = A² + B² + 2AB. b. (IR - A)(In + A) = In - A2. C. A + B is invertible.

d. (AB)-1 = A-B-1
e. A is invertible.

Answer 1

a. False, (A + B)² = A² + 2AB + B². b. False, (IR - A)(In + A) = IR + IRA - AIn - A². c. True, A + B is invertible. d. False, (AB)-1 = B-1A-1. e. True, A is invertible.

Step-by-step explanation:

a. False. The correct expansion and simplification of (A + B)² is (A + B)(A + B) = A(A + B) + B(A + B) = A² + AB + BA + B² = A² + 2AB + B².

b. False. The correct expansion and simplification of (IR - A)(In + A) is (IR - A)(In + A) = IR(In + A) - A(In + A) = IRIn + IRA - AIn - AA = IR + IRA - AIn - A².

c. True. The sum of two invertible matrices A and B is also invertible.

d. False. The correct expression for the inverse of the product AB is (AB)-1 = B-1A-1.

e. True. A matrix A is invertible if and only if its determinant is nonzero.