To analyze the given matrices $M$ and $M^{-1}$, let's observe their properties.

Matrices:

1. $M^{-1}$: \begin{bmatrix} \frac{1}{1-k^2} & -\frac{k}{1-k^2} & 0 \\ -\frac{k}{1-k^2} & \frac{1}{1-k^2} & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
2. $M$: \begin{bmatrix} 1 & k & 0 \\ k & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Notes:

• $M^{-1}$ is the inverse of $M$, meaning $M \times M^{-1} = I$, where $I$ is the identity matrix.
• The determinant of $M$ must not be zero, as a non-invertible matrix does not have an inverse. For $M$ to have an inverse: $\det(M) = 1 - k^2 \neq 0 \implies k \neq \pm 1.$

Verification:

To confirm that $M^{-1}$ is indeed the inverse of $M$, calculate:

\begin{bmatrix} 1 & k & 0 \\ k & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} \frac{1}{1-k^2} & -\frac{k}{1-k^2} & 0 \\ -\frac{k}{1-k^2} & \frac{1}{1-k^2} & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$ Multiply the matrices step by step. Let me know if you would like me to explicitly carry out the multiplication to verify this relationship. --- Would you like a detailed verification of this calculation or further assistance on related topics? ### Related Questions: 1. What is the determinant of $$M$$, and how does it affect invertibility? 2. How do you prove matrix inversion properties? 3. What are the eigenvalues and eigenvectors of $$M$$? 4. Can $$M^{-1}$$ be expressed in terms of elementary row operations on $$M$$? 5. What are the conditions under which a 3x3 matrix is invertible? ### Tip: When verifying inverses, always check both $$M \times M^{-1} = I$$ and $$M^{-1} \times M = I$$ to ensure correctness.