https://file.aiquickdraw.com/mathbot/file/172508212717039gw9165.jpg

The determinant of the matrix can be computed using the method of the principal diagonal (also known as the Laplace expansion along the diagonal).

The matrix given is:
\[
\begin{vmatrix}
0 & 9 & -3 & 0 & -4 \\
-5 & 0 & 7 & 5 & 5 \\
-2 & 0 & -10 & 3 & -7 \\
0 & 0 & -10 & 0 & 4 \\
-8 & 0 & -7 & 1 & 0
\end{vmatrix}
\]

To calculate the determinant using the principal diagonal method, you sum the products of the diagonals starting from the top left to bottom right and subtract the sum of the products of the diagonals starting from the top right to bottom left.

### Step 1: Identifying the diagonals
**Main diagonals (from top left to bottom right):**

1. \(0 \times 0 \times -10 \times 0 \times 0 = 0\)
2. \(9 \times 7 \times 3 \times 4 \times -8 = -6048\)
3. \(-3 \times 5 \times -7 \times 0 \times 0 = 0\)

**Secondary diagonals (from top right to bottom left):**

1. \(-4 \times 5 \times -10 \times 0 \times -8 = 0\)
2. \(0 \times 7 \times -7 \times 4 \times 0 = 0\)
3. \(9 \times 7 \times 3 \times -10 \times -8 = 6048\)

### Step 2: Calculation
To find the determinant \(D\), you subtract the sum of the secondary diagonals from the sum of the main diagonals:

\[
D = \text{Sum of main diagonals} - \text{Sum of secondary diagonals}
\]

\[
D = (-6048 + 0 + 0) - (0 + 0 + 6048) = -6048 - 6048 = -12096
\]

### Conclusion:
The determinant of the matrix is \(-12096\).

Would you like further details or have any questions about this method?

Here are some related questions:
1. How does the cofactor expansion method work for calculating determinants?
2. What is the significance of a matrix's determinant in linear algebra?
3. Can a matrix have a determinant of zero? What does it imply?
4. How would the determinant change if one row of the matrix was multiplied by a scalar?
5. How can the determinant help in solving systems of linear equations?

**Tip:** Always verify your determinant calculations by checking for zeros and symmetries in the matrix, which can simplify your work.

Learn how to calculate the determinant of a matrix using the principal diagonal method. Step-by-step solution with detailed explanation and example matrix provided. Suitable for undergraduate students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation