1.Reduce to triangular form [[3, - 4, - 5], [- 9, 1, 4], [- 5, 3, 1]]

To reduce the matrix 

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

to triangular form, we can use Gaussian elimination. Here are the steps:

1. **Start with the first row**: We can use the first row to eliminate the leading coefficients below it.

2. **Eliminate the first element of the second row**:
   - Multiply the first row by 3 and add it to the second row:
   \[
   R_2 = R_2 + 3R_1 \rightarrow [-9 + 3 \cdot 3, 1 + 3 \cdot (-4), 4 + 3 \cdot (-5)] = [0, -11, -11]
   \]

3. **Eliminate the first element of the third row**:
   - Multiply the first row by \(\frac{5}{3}\) and add it to the third row:
   \[
   R_3 = R_3 + \frac{5}{3}R_1 \rightarrow [-5 + \frac{5}{3} \cdot 3, 3 + \frac{5}{3} \cdot (-4), 1 + \frac{5}{3} \cdot (-5)] = [0, -\frac{11}{3}, -\frac{14}{3}]
   \]

The matrix now looks like this:
\[
\begin{bmatrix}
3 & -4 & -5 \\
0 & -11 & -11 \\
0 & -\frac{11}{3} & -\frac{14}{3}
\end{bmatrix}
\]

4. **Eliminate the leading coefficient in the third row**:
   - Multiply the second row by \(\frac{1}{11}\) and use it to eliminate the leading coefficient of the third row:
   \[
   R_3 = R_3 + \frac{1}{3}R_2 \rightarrow [0, -\frac{11}{3} + \frac{1}{3}(-11), -\frac{14}{3} + \frac{1}{3}(-11)] = [0, 0, -7]
   \]

The final triangular form of the matrix is:
\[
\begin{bmatrix}
3 & -4 & -5 \\
0 & -11 & -11 \\
0 & 0 & -7
\end{bmatrix}
\]

If you need further details or have questions about any steps, feel free to ask! Here are some related questions you might consider:

1. What is the significance of triangular form in linear algebra?
2. How can we use triangular matrices to solve systems of equations?
3. What is the next step after achieving triangular form if we want to find the solution to the equations?
4. How does the Gaussian elimination method compare to other methods for solving linear systems?
5. Can you provide a practical application of triangular matrices in real-world problems?

**Tip**: Always check your row operations for accuracy to avoid mistakes in calculations!

Reduce to triangular form [[3, - 4, - 5], [- 9, 1, 4], [- 5, 3, 1]]

Explore the process of reducing the matrix [[3, -4, -5], [-9, 1, 4], [-5, 3, 1]] to triangular form using Gaussian elimination. This detailed guide covers essential linear algebra concepts and step-by-step row operations suitable for high school students.

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