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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.

Write the correct properties of invertible matrices. a. If A is an invertible matrix, then (A^{-1})^{-1} = ?. b. If A is an invertible matrix and c is a nonzero scalar, then (cA)^{-1} = ?. c. If A and B are invertible matrices of the same size, then (AB)^{-1} = ?. d. If A is an invertible matrix, then (A^T)^{-1} = ?.

Explore the properties of invertible matrices with detailed solutions for inverse, scalar multiplication, matrix multiplication, and transpose operations. This problem involves core concepts in matrix algebra suitable for advanced high school or undergraduate students.

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