https://file.aiquickdraw.com/mathbot/file/17297145052974y3esb6k.jpg

Let's analyze the image and determine how to calculate the determinant of the matrix \( A \).

### Gauss Elimination and the Matrix

The sequence shown in the image represents the Gaussian elimination steps performed on matrix \( A \). The elementary matrices \( E_1 \), \( E_2 \), \( E_3 \), and \( E_4 \) transform matrix \( A \) into a new form. Eventually, the final matrix is:

\[
E_4 E_3 E_2 E_1 A = \begin{bmatrix} -9 & -9 & 1 \\ 0 & 6 & -5 \\ 0 & 0 & -4 \end{bmatrix}
\]

This matrix is upper triangular, which means that its determinant is the product of its diagonal entries:

\[
\text{det}(E_4 E_3 E_2 E_1 A) = (-9)(6)(-4) = 216
\]

### Impact of Elementary Matrices

Each elementary matrix \( E_i \) represents a row operation. There are three types of elementary row operations:

1. **Row swaps**: These change the sign of the determinant.
2. **Row scaling**: This multiplies the determinant by a scaling factor.
3. **Row addition**: This does not affect the determinant.

Looking at the transformations, it is likely that some scaling or row swaps occurred. The matrix \( A \) transforms into the final triangular matrix. However, since we don't know the exact operations from the image, we assume that the determinant is calculated from the final matrix.

Therefore, the determinant of \( A \) is \( 216 \), assuming no row swaps (which would change the sign).

Would you like more details or have questions about the steps?

### Related Questions:

1. How do elementary matrices affect the determinant?
2. How is the determinant of a triangular matrix calculated?
3. What are the types of row operations in Gaussian elimination?
4. Why does a row swap change the sign of the determinant?
5. How can Gaussian elimination be used to solve a system of linear equations?

**Tip**: When working with Gaussian elimination, always track how row operations affect the determinant. Row swaps change the sign, and scaling rows changes the magnitude.

What is the determinant of matrix A, given the transformations shown through Gaussian elimination?

This problem demonstrates how Gaussian elimination transforms a matrix step by step. Learn how to find the determinant by examining the resulting upper triangular matrix, considering the effects of row operations and elementary matrices. Suitable for linear algebra students.

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