Based on the matrix below, find the following:
𝐴 = | 1   2 −1 |
      | 2  1   2  |
      |−1  2  1  |

a) Determinant of the Matrix

To find the determinant of the \( 3 \times 3 \) matrix \( A \), we can use the following formula:

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

For the given matrix \( A \):

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

Here, \( a = 1 \), \( b = 2 \), \( c = -1 \), \( d = 2 \), \( e = 1 \), \( f = 2 \), \( g = -1 \), \( h = 2 \), \( i = 1 \).

Substituting these values into the formula:

\[
\text{det}(A) = 1(1 \cdot 1 - 2 \cdot 2) - 2(2 \cdot 1 - 2 \cdot -1) + (-1)(2 \cdot 2 - 1 \cdot -1)
\]

Let's calculate each part:

\[
\text{det}(A) = 1(1 - 4) - 2(2 + 2) + (-1)(4 + 1)
\]

\[
\text{det}(A) = 1(-3) - 2(4) + (-1)(5)
\]

\[
\text{det}(A) = -3 - 8 - 5 = -16
\]

So, the determinant of matrix \( A \) is \( \boxed{-16} \).

Would you like more details or have any other questions?

Here are 8 follow-up questions that you might find helpful:

1. How does the determinant of a matrix affect its invertibility?
2. What are the properties of determinants that are useful in matrix operations?
3. How does the determinant relate to the volume of a parallelepiped in geometry?
4. What happens to the determinant if two rows of a matrix are swapped?
5. How can the determinant be used to solve systems of linear equations?
6. What is the significance of a zero determinant?
7. How do you compute the determinant of a larger \( n \times n \) matrix?
8. Can the determinant be used to find eigenvalues of a matrix?

**Tip:** Always remember that the determinant of a matrix provides important insights into the properties of the matrix, including whether it is invertible and the scaling factor of the linear transformation represented by the matrix.

Learn how to calculate the determinant of matrix A = | 1 2 -1; 2 1 2; -1 2 1 | with step-by-step instructions and detailed formula application. This problem covers matrix algebra concepts and determinant calculations, suitable for Grades 10-12 students.

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