Question

Select all statements below which are true for all invertible n×n matrices A and B

A. (A+B)²=A²+B²+2AB
B. A+A⁻¹ is invertible
C. (In+A)(In+A⁻¹)=2In+A+A⁻¹
D. A⁶ is invertible
E. (A+A⁻¹)⁹=A⁹+A⁹
F. AB=BA

Answer 1

The true statements for all invertible n×n matrices A and B are: (A+B)²=A²+B²+2AB, A+A⁻¹ is invertible, (In+A)(In+A⁻¹)=2In+A+A⁻¹, and AB=BA.

Step-by-step explanation:

The following statements are true for all invertible n×n matrices A and B:

1. Statement A: (A+B)²=A²+B²+2AB
2. Statement B: A+A⁻¹ is invertible
3. Statement C: (In+A)(In+A⁻¹)=2In+A+A⁻¹
4. Statement D: A⁶ is invertible
5. Statement E: (A+A⁻¹)⁹=A⁹+A⁹
6. Statement F: AB=BA

Step-by-step explanation:

1. Statement A is true because (A+B)² = A² + AB + BA + B² = A² + 2AB + B²
2. Statement B is true because A+A⁻¹ is the sum of an invertible matrix A and its inverse, which is always invertible.
3. Statement C is true because (In+A)(In+A⁻¹) = In(In+A⁻¹) + A(In+A⁻¹) = In+A + A⁻¹ + A + A⁻¹ = 2In + A + A⁻¹
4. Statement D is false because raising an invertible matrix to any power will always result in an invertible matrix.
5. Statement E is false because the powers of (A+A⁻¹) cannot be simplified in that way.
6. Statement F is true because the commutative property of matrix multiplication states that AB = BA.

From the above explanations, the correct statements are A, B, C, and F.