https://file.aiquickdraw.com/mathbot/file/1729841797123xfkwlc91.jpg

The matrix in the image is:

\[
\begin{bmatrix}
2 & 1 & -3 \\
1 & 1 & 0 \\
0 & 4 & -1
\end{bmatrix}
\]

Now let's compute the determinant of this 3x3 matrix.

The determinant \( \text{det}(A) \) for a 3x3 matrix:

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

is calculated as:

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

For the given matrix, we have:
\[
\text{det}(A) = 2 \cdot (1 \cdot -1 - 0 \cdot 4) - 1 \cdot (1 \cdot -1 - 0 \cdot 0) + (-3) \cdot (1 \cdot 4 - 1 \cdot 0)
\]
\[
= 2 \cdot (-1) - 1 \cdot (-1) + (-3) \cdot 4
\]
\[
= -2 + 1 - 12 = -13
\]

So, the determinant of the matrix is \( -13 \).

Would you like further details on this computation or have any other questions? Here are some related topics:

1. How is the determinant of a matrix used in solving systems of equations?
2. What are some properties of determinants in larger matrices?
3. How does the determinant change when rows or columns are swapped?
4. How does a determinant help in finding the inverse of a matrix?
5. What is the significance of a zero determinant?

**Tip:** A matrix with a zero determinant is called singular, meaning it does not have an inverse.

What is the determinant of the given matrix?

Learn how to calculate the determinant of a 3x3 matrix using the standard determinant formula. This example walks through the steps using a specific matrix and includes detailed explanations of the calculations. Suitable for Grades 10-12.

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