Matrix True And False Statements

Hi I just wanted to double check the answers for these T/F Matrix statements, hopefully they are simple to determine for you guys...

Which of the following statements are true for all *A* and *B* invertible *n*×*n* matrices?

**1.** *A**B*=*I* does not imply that *B**A*=*I*.

**2.** The columns of *B* span R*n*.

**3.** *B* can have two identical rows.

**4.** The homogeneous system *B***v**=**0** has only the trivial solution.

**5.** The matrix *A**T* is invertible.

**6.** (*A**B**A^*−1)^3=*A**B^*3*A^*−1

**7.** *A^*7*B^*5 is invertible

**8.** (*A**B*)^−1=*A^*−1*B^*−1

**9.** *A*+*A^*−1 is invertible

**10. **(*A*+*B*)(*A*−*B*)=*A^*2−*B^*2

Thank you!

Re: Matrix True And False Statements

What is your sequence of answers?

Re: Matrix True And False Statements

1. T

2. F

3. T

4. F

5. T

6. F

7.t

8. T

9. F

10. F

Re: Matrix True And False Statements

Some of the statements are unclear to me.

Let me start with 1.

AB=I does not imply that BA=I. Since A is invertible inv(A)AB = inv(A)I or B = inv(A). Which means that A inv(A)=I does not imply that inv(A) A=I. The first statement is thus false.

I think you should reconsider your answers. The binomial inverse theorem may help. Binomial inverse theorem - Wikipedia, the free encyclopedia