https://file.aiquickdraw.com/mathbot/file/1737373665275evwxzogj.jpg

The matrix you provided is a 3x3 matrix:

\[
\begin{pmatrix}
-5 & -5 & 0 \\
-3 & 3 & -5 \\
5 & 6 & -3
\end{pmatrix}
\]

To calculate the determinant of this matrix, we can use the formula for a 3x3 determinant:

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

For matrix:

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

Substituting the values from the matrix:

\[
\text{det}(A) = (-5)((3)(-3) - (-5)(6)) - (-5)((-3)(-3) - (-5)(5)) + (0)((-3)(6) - (3)(5))
\]

Simplifying step-by-step:

\[
= (-5)((-9) + 30) - (-5)((9) + 25) + 0
\]
\[
= (-5)(21) + 5(34)
\]
\[
= -105 + 170
\]
\[
= 65
\]

Therefore, the determinant of the matrix is **65**.

Would you like a further explanation or details? Here are some related questions you can explore:

1. How to calculate the determinant of a 2x2 matrix?
2. What are the properties of a determinant?
3. How does the determinant of a matrix affect its invertibility?
4. What does a zero determinant imply about a matrix?
5. Can a matrix with all elements zero still have a non-zero determinant?

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

Calculate the determinant of the matrix: [[-5, -5, 0], [-3, 3, -5], [5, 6, -3]].

Learn how to calculate the determinant of a 3x3 matrix using cofactor expansion. This problem involves the matrix [[-5, -5, 0], [-3, 3, -5], [5, 6, -3]]. Suitable for high school students in Grades 9-11 studying linear algebra.

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