https://file.aiquickdraw.com/mathbot/file/17268255967128hy255od.jpg

We are asked to find the determinant of matrix \( A \) using cofactor expansion along any row or column. The matrix \( A \) is:

\[
A = \begin{bmatrix}
-5 & 0 & 5 \\
5 & 7 & 1 \\
-1 & 0 & 6
\end{bmatrix}
\]

I will perform the cofactor expansion along the first row, as it contains zeros, which simplifies the computation.

The formula for the cofactor expansion along the first row is:

\[
\text{det}(A) = (-5) \cdot \text{det}\begin{bmatrix} 7 & 1 \\ 0 & 6 \end{bmatrix} - 0 \cdot \text{det} \begin{bmatrix} 5 & 1 \\ -1 & 6 \end{bmatrix} + 5 \cdot \text{det} \begin{bmatrix} 5 & 7 \\ -1 & 0 \end{bmatrix}
\]

### Step 1: Calculate \( \text{det} \begin{bmatrix} 7 & 1 \\ 0 & 6 \end{bmatrix} \)
\[
\text{det} \begin{bmatrix} 7 & 1 \\ 0 & 6 \end{bmatrix} = (7 \cdot 6) - (1 \cdot 0) = 42
\]

### Step 2: Calculate \( \text{det} \begin{bmatrix} 5 & 7 \\ -1 & 0 \end{bmatrix} \)
\[
\text{det} \begin{bmatrix} 5 & 7 \\ -1 & 0 \end{bmatrix} = (5 \cdot 0) - (7 \cdot -1) = 7
\]

### Step 3: Substitute the values into the cofactor expansion formula:
\[
\text{det}(A) = (-5)(42) + 0 + 5(7)
\]
\[
\text{det}(A) = -210 + 0 + 35 = -175
\]

Thus, the determinant of the matrix \( A \) is:

\[
\boxed{-175}
\]

Would you like more details or have any questions?

Here are five related questions you might consider:
1. How is the cofactor expansion method applied to a 4x4 matrix?
2. What is the significance of choosing a row or column with many zeros for cofactor expansion?
3. Can the cofactor expansion method be used for matrices larger than 3x3?
4. How do you find the determinant of a 2x2 matrix?
5. What is the geometric interpretation of a matrix's determinant?

**Tip:** When using cofactor expansion, it's always a good idea to pick a row or column with as many zeros as possible to reduce calculation complexity.

Evaluate det(A) by a cofactor expansion along a row or column of your choice. Matrix A = [[-5, 0, 5], [5, 7, 1], [-1, 0, 6]]

Step-by-step guide on how to evaluate the determinant of a 3x3 matrix using cofactor expansion. Learn how to break down the matrix, apply cofactor expansion, and calculate smaller 2x2 matrix determinants to find the result. Suitable for undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation