https://file.aiquickdraw.com/mathbot/file/1726948409269a491tmpj.jpg

The image shows a \(3 \times 3\) matrix \( A \) and asks for the determinant of this matrix. Matrix \( A \) is given as:

\[
A = \begin{bmatrix}
-6 & -4 & 3 \\
0 & -2 & 8 \\
0 & 0 & -7
\end{bmatrix}
\]

To calculate the determinant of matrix \( A \), we can observe that it is an upper triangular matrix, meaning all the elements below the diagonal are zero. The determinant of an upper triangular matrix is simply the product of the diagonal elements.

So, we compute the determinant:

\[
\det(A) = (-6) \times (-2) \times (-7)
\]

Let's calculate this.The determinant of matrix \( A \) is:

\[
\det(A) = -84
\]

Would you like more details or have any further questions?

Here are some related questions you might explore:
1. How do you calculate the determinant of a non-triangular \( 3 \times 3 \) matrix?
2. What are the properties of upper and lower triangular matrices?
3. How is the determinant of a matrix used in solving linear equations?
4. Can you find the inverse of a matrix with determinant \( -84 \)?
5. What does the sign of the determinant tell us about the matrix?

**Tip:** For square matrices, the determinant provides key information about invertibility: a non-zero determinant means the matrix is invertible.

What is the determinant of the matrix A = [[-6, -4, 3], [0, -2, 8], [0, 0, -7]]?

Learn how to calculate the determinant of the 3x3 upper triangular matrix A = [[-6, -4, 3], [0, -2, 8], [0, 0, -7]]. This problem involves the properties of triangular matrices and is suitable for students in Grades 10-12.

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