Problem Breakdown:

We are tasked to verify/discuss the validity of the two properties related to the inverse of matrices and scalar multiplication:

Given:

• $A$ and $B$ are $n \times n$ nonsingular matrices ($\text{det}(A) \neq 0$, $\text{det}(B) \neq 0$),
• $c$ is any real number ($c \neq 0$),

Properties:

1. $(AB)^{-1} = B^{-1}A^{-1}$
2. $(cA)^{-1} = \frac{1}{c}A^{-1}$

Property 1: $(AB)^{-1} = B^{-1}A^{-1}$

Proof:

By definition of the inverse of a matrix, the product of a matrix and its inverse yields the identity matrix $I$: $(AB)(AB)^{-1} = I.$

Now expand $AB$ as a product of two matrices: $AB \cdot (AB)^{-1} = I.$

Suppose $(AB)^{-1} = B^{-1}A^{-1}$, then substituting this expression: $AB \cdot (B^{-1}A^{-1}) = I.$

Using the associative property of matrix multiplication: $(A \cdot B) \cdot (B^{-1} \cdot A^{-1}) = A \cdot (B \cdot B^{-1}) \cdot A^{-1}.$

Since $B \cdot B^{-1} = I$, this simplifies to: $A \cdot I \cdot A^{-1} = A \cdot A^{-1}.$

Finally, $A \cdot A^{-1} = I$, so the equation holds: $I = I.$

Thus, Property 1 is valid: $(AB)^{-1} = B^{-1}A^{-1}.$

Property 2: $(cA)^{-1} = \frac{1}{c}A^{-1}$

Proof:

By definition of scalar multiplication and matrix inversion: $(cA)(cA)^{-1} = I.$

Assume $(cA)^{-1} = \frac{1}{c}A^{-1}$, then substitute: $(cA) \cdot \left(\frac{1}{c}A^{-1}\right) = I.$

Distribute $c$ across the product: $c \cdot (A \cdot \frac{1}{c}A^{-1}) = I.$

Factor $c$ and $\frac{1}{c}$: $c \cdot \frac{1}{c} \cdot (A \cdot A^{-1}) = I.$

Since $c \cdot \frac{1}{c} = 1$ and $A \cdot A^{-1} = I$, this simplifies to: $1 \cdot I = I.$

Thus, Property 2 is valid: $(cA)^{-1} = \frac{1}{c}A^{-1}.$

Final Results:

• Property 1: $(AB)^{-1} = B^{-1}A^{-1}$ is true.
• Property 2: $(cA)^{-1} = \frac{1}{c}A^{-1}$ is true.

Any questions or clarifications needed?

Follow-Up Questions:

1. Why does the order of matrix multiplication matter for inverses?
2. Can we generalize the first property to more than two matrices?
3. What happens to the inverses if $c = 0$?
4. Why is matrix nonsingularity essential in these proofs?
5. How can these properties be applied to solving linear equations?

Tip:

Always check the determinant of a matrix or scalar before calculating its inverse; the determinant must be nonzero for the inverse to exist.