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The given problem asks to find the values of \( k \) that make the matrix \( A \) invertible. The matrix \( A \) is:

\[
A = \begin{pmatrix} 
1 & 1 & k \\
1 & 2 & 7 \\
1 & 3 & 1 
\end{pmatrix}
\]

To determine when a matrix is invertible, we need to check when its determinant is non-zero. The determinant of a \( 3 \times 3 \) matrix is given by:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

Where the matrix is:

\[
\begin{pmatrix} 
a & b & c \\
d & e & f \\
g & h & i 
\end{pmatrix}
\]

For matrix \( A \), we have:

\[
\begin{pmatrix} 
1 & 1 & k \\
1 & 2 & 7 \\
1 & 3 & 1 
\end{pmatrix}
\]

Thus:
- \( a = 1 \)
- \( b = 1 \)
- \( c = k \)
- \( d = 1 \)
- \( e = 2 \)
- \( f = 7 \)
- \( g = 1 \)
- \( h = 3 \)
- \( i = 1 \)

Now, let's calculate the determinant. I'll do this calculation.The determinant of matrix \( A \) is \( \text{det}(A) = k - 13 \).

For the matrix \( A \) to be invertible, its determinant must be non-zero. Therefore, \( k - 13 \neq 0 \), which gives:

\[
k \neq 13
\]

Thus, the matrix \( A \) is invertible for all values of \( k \) except \( k = 13 \).

Would you like more details or have any questions about this?

Here are 5 related questions you could explore:
1. What happens to the matrix when \( k = 13 \)?
2. How do you calculate the inverse of a \( 3 \times 3 \) matrix?
3. How does the determinant help in determining the invertibility of a matrix?
4. What are some real-world applications of matrix invertibility?
5. How does row reduction relate to determining if a matrix is invertible?

**Tip:** A matrix with a determinant of zero is called "singular" and cannot be inverted.

Considere a seguinte matriz A = [[1, 1, k], [1, 2, 7], [1, 3, 1]]. Determine todos os valores de k que tornam a matriz A invertível.

Solve for k to determine when the matrix A = [[1, 1, k], [1, 2, 7], [1, 3, 1]] is invertible by calculating its determinant. This problem involves matrix theory and is suitable for undergraduate students.

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