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Select all statements below which are true for all invertible $n\times n$ matrices $A$ and $B$

A. $(A+A^{−1})^4=A^4+A^{−4}$ (true)

B. $A+I_n$ is invertible (False)

C. $A^7 B^2$ is invertible (true)

D. $AB=BA$ (False)

E. $(A+B)^2=A^2+B^2+2AB$ (False)

F. $(I_n−A)(I_n+A)=I_n−A^2$ (true)

Rodrigo de Azevedo
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Mirzak Haraki
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    Please see https://math.meta.stackexchange.com/questions/5020/ Does A. say $(A+A^{-1})^4=A^4+A^{-4}$? – Angina Seng Oct 20 '19 at 17:30
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    This is hard to read. Can you explain why the first statement is true for, say, the identity matrix? I expect this just comes down to explaining what the first statement actually says. – lulu Oct 20 '19 at 17:32
  • I think the first statement says <<$(A + A^{-1})^4 = A^4 + A^{-4}$ is true>> and the OP asks if this is correct. So the answer is "no, the first statement is incorrect." – WhatsUp Oct 20 '19 at 17:43
  • @WhatsUp, what's the OP? –  Oct 20 '19 at 17:47
  • the content between parentheses is the one that I put. For example I think that the first one is correct so I added "true". – Mirzak Haraki Oct 20 '19 at 17:57
  • The OP is the original poster – Mirzak Haraki Oct 20 '19 at 17:57

2 Answers2

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Statement A is false. The inverse of a matrix $A$ is defined such that $A^{-1}A=AA^{-1}=I$, where $I$ is the identity matrix. We can see that $(A+A^{-1})^4=A^4+A^{-4})$ by simply expanding $(A+A^{-1})^4$ using the binomial theorem. Doing so and using the above identity, we have $(A+A^{-1})^4 = A^4 + 4(A^2+A^{-2})+6I+A^{-4}\neq A^4+A^{-4}$. Take the $2\times 2$ invertible matrix $M= \left[ {\begin{array}{} 2 & 0\\ 0 & 2\\ \end{array} } \right] $ as a counterexample.

Statement B is false. Use the matrix $-M$ and choose $n=2$. The resulting determinant will be $0$.

Statement C is true. Simply take the matrix $B^{-2}A^{-7}$ and by the properties of matrix inverses and the identity matrix, you'll get the identity matrix.

Statement D is false. Matrix multiplication is known not to be commutative and it is easy to see this if you take any two assymmetric matrices (i.e. it is not equal to its transpose).

Statement E is also false. Using the fact that statement D is false and performing the specified multiplication, we obtain $(A+B)^2 = A^2 + AB + BA + B^2 \neq A^2 + 2AB + B^2$ because it is not true that $AB= BA$ for all $n\times n$ matrices $A$ and $B$.

Statement F is false. It should read $(In-A)(In+A)=In^2-A^2$ since $n$ is an integer. This is due to the properties of matrix multiplication and the fact that $I^2 = I$.

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$(A+A^{-1})^4 = A^4 + 4A^2 + 6I + 4A^{-2}+ A^{-4}$

$(nI + A)(nI-A) = n^2I - A^2$

Doug M
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