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To determine when the given matrix is invertible, we need to find for which values of \( k \) the determinant of the matrix is non-zero.

The matrix given is:
\[
\begin{pmatrix}
2 & 5 & 1 \\
0 & k & -3 \\
0 & 3 & 9
\end{pmatrix}
\]

The determinant of a 3x3 matrix can be found using cofactor expansion along the first row. For this matrix:

\[
\text{determinant} = 2 \cdot \text{det}\begin{pmatrix} k & -3 \\ 3 & 9 \end{pmatrix} - 5 \cdot \text{det}\begin{pmatrix} 0 & -3 \\ 0 & 9 \end{pmatrix} + 1 \cdot \text{det}\begin{pmatrix} 0 & k \\ 0 & 3 \end{pmatrix}
\]

Now, let's calculate each 2x2 determinant:

1. \(\text{det}\begin{pmatrix} k & -3 \\ 3 & 9 \end{pmatrix} = k(9) - (-3)(3) = 9k + 9\)
2. \(\text{det}\begin{pmatrix} 0 & -3 \\ 0 & 9 \end{pmatrix} = 0\) (since the first column has all zeros)
3. \(\text{det}\begin{pmatrix} 0 & k \\ 0 & 3 \end{pmatrix} = 0\) (same reason, first column is zeros)

Thus, the determinant becomes:

\[
\text{determinant} = 2 \cdot (9k + 9) - 5 \cdot 0 + 1 \cdot 0 = 18k + 18
\]

For the matrix to be invertible, the determinant must be non-zero:

\[
18k + 18 \neq 0
\]

Solving for \( k \):

\[
18k \neq -18 \quad \Rightarrow \quad k \neq -1
\]

Therefore, the matrix is invertible for all \( k \neq -1 \).

### Correct answer: None of the provided options directly match \( k \neq -1 \), but **option F**: \( k \neq -2 \) might be a misprint. **The correct condition is \( k \neq -1 \)**.

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Do you want further details or have any questions?

Here are 5 related questions to consider:
1. How is the determinant of a 3x3 matrix calculated?
2. What is the significance of a matrix having a determinant of zero?
3. How do cofactor expansions work for higher-dimensional matrices?
4. How does the invertibility of a matrix relate to its eigenvalues?
5. What are the applications of matrix invertibility in solving systems of equations?

**Tip**: Always double-check the conditions of invertibility by calculating the determinant carefully to avoid errors due to zeros in specific matrix rows or columns.

Find all values of k for which the matrix is invertible? A: k=0, B: Any k ≠ 0, C: k=3, D: Any k ≠ 3, E: k = -2, F: Any k ≠ -2, G: Any k ∈ R.

This problem involves finding the values of k for which a 3x3 matrix is invertible by calculating the determinant and solving for k using cofactor expansion. Learn key concepts related to matrix invertibility and determinants.

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