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To reduce the matrix 

\[
\begin{bmatrix} 3 & -4 & -5 \\ -9 & 1 & 4 \\ -5 & 3 & 1 \end{bmatrix}
\]

to triangular form, we need to perform Gaussian elimination to get zeros below the leading entries in each row. Here’s the step-by-step process:

1. **Row Operations**: We'll perform row operations to make the elements below the first entry (3 in the top left corner) equal to zero.

### Step-by-Step Solution

1. **Eliminate** the entry below the 3 in the first column.
   - Use row 1 as the pivot row.
   - To eliminate \(-9\) in row 2, calculate \( R_2 + 3R_1 \):
     \[
     R_2 = R_2 + 3 \times R_1 = [-9 + 3 \times 3, 1 + 3 \times (-4), 4 + 3 \times (-5)] = [0, -11, -11]
     \]
   - To eliminate \(-5\) in row 3, calculate \( R_3 + \frac{5}{3}R_1 \):
     \[
     R_3 = R_3 + \frac{5}{3} \times R_1 = [-5 + 5,

Reduce to triangular form the matrix:

[ 3  -4  -5 ]
[ -9  1   4 ]
[ -5  3   1 ]

Learn how to reduce a 3x3 matrix to triangular form using Gaussian elimination. This step-by-step guide covers the key concepts of linear algebra and matrix row operations, ideal for students in Grades 11-12 or undergraduate courses.

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