The image asks for filling in the correct properties of invertible matrices. Let’s go through each statement and provide the correct answers:

a. If $A$ is an invertible matrix, then $A^{-1}$ is invertible, and $(A^{-1})^{-1} = A$.

b. If $A$ is an invertible matrix and $c$ is a nonzero scalar, then $cA$ is an invertible matrix, and $(cA)^{-1} = \frac{1}{c}A^{-1}$.

c. If $A$ and $B$ are invertible matrices of the same size, then $AB$ is invertible, and $(AB)^{-1} = B^{-1}A^{-1}$.

d. If $A$ is an invertible matrix, then $A^T$ (the transpose of $A$) is invertible, and $(A^T)^{-1} = (A^{-1})^T$.

Would you like further details on any of these properties?

Here are five related questions to further expand your understanding:

1. What are the conditions under which a matrix is invertible?
2. How does the determinant of a matrix relate to its invertibility?
3. Can the inverse of a non-square matrix exist?
4. How do these properties extend to block matrices?
5. What happens when you invert a matrix product involving more than two matrices?

Tip: Remember that matrix multiplication is generally not commutative, meaning $AB \neq BA$ in most cases, which is why the order of inverses matters.