Which Matrix Formulas Are Universally True for Invertible Matrices?

[mshiddensecret]

Homework Statement

Determine which of the formulas hold for all invertible nhttp://msr02.math.mcgill.ca/webwork2_files/jsMath/fonts/cmsy10/alpha/144/char02.png n matrices A andB

A. 7A is invertible
B. ABA^−1=B
C. A+B is invertible
D. (A+B)2=A2+B2+2AB
E. (A+A^−1)^8=A8+A−8
F. (ABA^−1)^3=AB3A−1

Homework Equations

The Attempt at a Solution

I think 1 should be true and c should be true. Those are for certain. The rest, I don't know how.

 

[Mark44]

Homework Statement

Determine which of the formulas hold for all invertible nhttp://msr02.math.mcgill.ca/webwork2_files/jsMath/fonts/cmsy10/alpha/144/char02.png n matrices A andB

A. 7A is invertible
B. ABA^−1=B
C. A+B is invertible
D. (A+B)2=A2+B2+2AB
E. (A+A^−1)^8=A8+A−8
F. (ABA^−1)^3=AB3A−1

Homework Equations

The Attempt at a Solution

I think 1 should be true and c should be true. Those are for certain. The rest, I don't know how.

Your efforts just barely qualify as an attempt at a solution...

For A (not 1), you are correct. Can you figure out what the inverse is?
For C, consider these matrices:
$$A = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$$
$$B = \begin{bmatrix} -1 & 0 \\ 0 & -1\end{bmatrix}$$
Both A and B are invertible (clearly, I hope). Is their sum invertible?

For the others, try expanding what's on the left side of the given equation, and see if you get what's on the right side.

Your book should have some properties or theorems of invertible matrices.

 

[mshiddensecret]

Still having trouble. B is true. C is false. D is false after expanding. Don't know how to expand e and f.

 

[Mark44]

For E you have "(A+A^−1)^8=A8+A−8"
Should the right side be A⁸ + A^-8?
At the very least, use ^ to indicate exponents.

For E, how do you expand a binomial?
For F, what properties do you know of to help with expanding ABA^-1 to the third power?

 

[mshiddensecret]

So after like 100x, I finally got the answer. Turns out A and F are the only true ones. I don't understand why though.

 

[Mark44]

So after like 100x, I finally got the answer. Turns out A and F are the only true ones. I don't understand why though.

Several of the problems test your understanding of matrix multiplication. In particular, that multiplication isn't commutative, so in general, AB $\neq$ BA.