Math Problem Statement

Reduce to triangular form the matrix: [ 3 -4 -5 ] [ -9 1 4 ] [ -5 3 1 ]

Solution

To reduce the matrix

[345914531]\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 9-9 in row 2, calculate R2+3R1R_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 5-5 in row 3, calculate R3+53R1R_3 + \frac{5}{3}R_1: [ R_3 = R_3 + \frac{5}{3} \times R_1 = [-5 + 5,

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Gaussian Elimination
Matrix Operations

Formulas

Gaussian elimination process
Row operations (R_i → R_i + k * R_j)

Theorems

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Suitable Grade Level

Undergraduate level or Grades 11-12