https://file.aiquickdraw.com/mathbot/file/1726829862270h0oj7uh4.jpg

I see an image showing a matrix with the task to evaluate its determinant using the arrow technique. The matrix given is:

\[
\text{det}\left( \begin{matrix} 
-1 & 2 & 3 \\
4 & 0 & -4 \\
1 & 5 & 2 
\end{matrix} \right)
\]

Let's calculate the determinant of this 3x3 matrix using the cofactor expansion method.

The determinant of a 3x3 matrix \(\text{det}(A)\) is calculated as:

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

For the matrix:

\[
A = \left( \begin{matrix} 
a & b & c \\
d & e & f \\
g & h & i 
\end{matrix} \right)
\]

Substituting the values from your matrix into the formula:

\[
a = -1, b = 2, c = 3
\]
\[
d = 4, e = 0, f = -4
\]
\[
g = 1, h = 5, i = 2
\]

I'll calculate the determinant now.The determinant of the matrix is:

\[
\text{det} = 16
\]

Would you like a more detailed explanation of the steps, or do you have any other questions?

Here are some related questions to further explore this topic:
1. What is the cofactor expansion method used in larger matrices?
2. How does the arrow method differ from other determinant calculation techniques?
3. Can the determinant of a matrix be used to solve systems of linear equations?
4. What does the determinant tell us about the invertibility of a matrix?
5. How can you interpret the geometric meaning of a determinant in 3D space?

**Tip:** When dealing with large matrices, applying properties like row reduction can simplify the determinant calculation.

Use the arrow technique to evaluate the determinant of the matrix:
[-1, 2, 3]
[4, 0, -4]
[1, 5, 2]

Learn how to calculate the determinant of a 3x3 matrix using the arrow technique. This example involves the matrix [-1, 2, 3; 4, 0, -4; 1, 5, 2], with a detailed step-by-step explanation of the cofactor expansion method. Suitable for high school students in Grades 10-12.

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