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The given matrix \( A \) is a \( 3 \times 3 \) matrix:

\[
A = \begin{pmatrix} 
2 & 1 & 91 \\
a & 1 & 2 \\
0 & 2 & 5 
\end{pmatrix}
\]

We are asked to compute the determinant of matrix \( A \). The formula for the determinant of a \( 3 \times 3 \) matrix is:

\[
\text{det}(A) = A_{11} \cdot \text{det}\begin{pmatrix} A_{22} & A_{23} \\ A_{32} & A_{33} \end{pmatrix} 
- A_{12} \cdot \text{det}\begin{pmatrix} A_{21} & A_{23} \\ A_{31} & A_{33} \end{pmatrix} 
+ A_{13} \cdot \text{det}\begin{pmatrix} A_{21} & A_{22} \\ A_{31} & A_{32} \end{pmatrix}
\]

Here, \( A_{ij} \) represents the element in the \( i \)-th row and \( j \)-th column of matrix \( A \). Let's compute the determinant using cofactor expansion:

### Step 1: Expansion along the first row
\[
\text{det}(A) = 2 \cdot \text{det} \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix} 
- 1 \cdot \text{det} \begin{pmatrix} a & 2 \\ 0 & 5 \end{pmatrix} 
+ 91 \cdot \text{det} \begin{pmatrix} a & 1 \\ 0 & 2 \end{pmatrix}
\]

### Step 2: Calculate the \( 2 \times 2 \) determinants
\[
\text{det} \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix} = (1)(5) - (2)(2) = 5 - 4 = 1
\]
\[
\text{det} \begin{pmatrix} a & 2 \\ 0 & 5 \end{pmatrix} = (a)(5) - (2)(0) = 5a
\]
\[
\text{det} \begin{pmatrix} a & 1 \\ 0 & 2 \end{pmatrix} = (a)(2) - (1)(0) = 2a
\]

### Step 3: Substitute back into the formula
\[
\text{det}(A) = 2 \cdot 1 - 1 \cdot 5a + 91 \cdot 2a
\]
\[
\text{det}(A) = 2 - 5a + 182a
\]
\[
\text{det}(A) = 2 + 177a
\]

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

\[
\boxed{2 + 177a}
\]

### Do you want details on any steps or have questions?

Here are some related questions:
1. How do you compute the cofactor expansion for larger matrices?
2. How does the determinant affect the invertibility of a matrix?
3. Can a matrix with a variable element in the determinant have multiple values for specific cases?
4. How do determinants relate to eigenvalues of a matrix?
5. How does row or column swapping affect the determinant?

**Tip:** The determinant of a matrix can often be simplified by choosing a row or column with many zeros for cofactor expansion, reducing calculation complexity.

Let A = [[2, 1, 91], [a, 1, 2], [0, 2, 5]]. Find the determinant of matrix A.

Learn how to compute the determinant of a 3x3 matrix that includes a variable. This problem utilizes the cofactor expansion method, suitable for high school students studying linear algebra concepts.

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