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The determinant given is for a 3x3 matrix \( A \), where the matrix is:

\[
A = \begin{pmatrix}
1 & -1 & 1 \\
1 & 2 & a \\
1 & 4 & 10
\end{pmatrix}
\]

To calculate the determinant \( \text{det}(A) \), we use the cofactor expansion along the first row:

\[
\text{det}(A) = 1 \cdot \begin{vmatrix} 2 & a \\ 4 & 10 \end{vmatrix}
- (-1) \cdot \begin{vmatrix} 1 & a \\ 1 & 10 \end{vmatrix}
+ 1 \cdot \begin{vmatrix} 1 & 2 \\ 1 & 4 \end{vmatrix}
\]

Let's calculate each of these 2x2 determinants:

1. \( \begin{vmatrix} 2 & a \\ 4 & 10 \end{vmatrix} = (2 \cdot 10) - (a \cdot 4) = 20 - 4a \)
2. \( \begin{vmatrix} 1 & a \\ 1 & 10 \end{vmatrix} = (1 \cdot 10) - (a \cdot 1) = 10 - a \)
3. \( \begin{vmatrix} 1 & 2 \\ 1 & 4 \end{vmatrix} = (1 \cdot 4) - (1 \cdot 2) = 4 - 2 = 2 \)

Now, substituting these into the original formula:

\[
\text{det}(A) = 1(20 - 4a) + 1(10 - a) + 1 \cdot 2
\]
\[
\text{det}(A) = 20 - 4a + 10 - a + 2
\]
\[
\text{det}(A) = 32 - 5a
\]

Thus, the determinant of matrix \( A \) is:

\[
\text{det}(A) = 32 - 5a
\]

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. What is the determinant of matrix \( A \) if \( a = 2 \)?
2. How does the value of \( a \) affect the determinant of this matrix?
3. Can you generalize the formula for the determinant of any 3x3 matrix?
4. How would the determinant change if you swapped rows in the matrix?
5. How does the determinant relate to whether a matrix is invertible?

**Tip:** The determinant can help determine if a matrix is invertible. If the determinant is zero, the matrix is not invertible!

Find the determinant of matrix A, where A = [[1, -1, 1], [1, 2, a], [1, 4, 10]]

Learn how to calculate the determinant of a 3x3 matrix A = [[1, -1, 1], [1, 2, a], [1, 4, 10]] using the cofactor expansion method. The step-by-step solution includes determining the 2x2 minors and simplifying to find det(A) = 32 - 5a. Ideal for undergraduate students studying Linear Algebra.

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