Discover Linear Algebra: A first course in linear algebra

Explain the meaning of the notation ${(M^{-1})}^{-1}\text{,}$ and give an explanation based on the definition of inverse of a matrix for why this formula must always compute to $M\text{.}$

Explain the meaning of the notation ${(MN)}^{-1}\text{,}$ and give an explanation based on the definition of inverse of a matrix for why the result of this formula is (in general) equal to $N^{-1}{M}^{-1}$ and not $M^{-1}{N}^{-1}\text{.}$

Let $\ell$ represent an arbitrary positive integer. Explain the difference in the meaning of the notation ${(M^{\ell })}^{-1}$ versus the notation ${(M^{-1})}^{\ell }\text{,}$ and give an explanation based on the definition of inverse of a matrix for why the two formulas always compute to the same result.

For each of Task a and Task b: if your answer is “sometimes,” under what conditions is it valid?

Explain what conclusions can be drawn about the number of solutions to the system from the assumption that $A$ is invertible.

What further can be said about the solution(s) in the case that the system is homogeneous (that is, $\mathbf{b}=\mathbf{0}$)?