Question

What three elimination matrices e21 e31 e32 put a into its upper triangular form

Answer 1

To transform matrix A into its upper triangular form using elimination matrices
`\(e_(21)\), \(e_(31)\)`, and
`\(e_(32)\)`, we perform a sequence of row operations.

1.
`\(e_(21)\)` eliminates the second row's first element:

`\[e_(21) \cdot A = \begin{bmatrix} 1 & 0 & 0 \\ -a_(21)/a_(11) & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \cdot A\]`

2.
`\(e_(31)\)` eliminates the third row's first element:

`\[e_(31) \cdot e_(21) \cdot A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -a_(31)/a_(11) & 0 & 1 \end{bmatrix} \cdot e_(21) \cdot A\]`

3.
`\(e_(32)\)` eliminates the third row's second element:

`\[e_(32) \cdot e_(31) \cdot e_(21) \cdot A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -a_(32)/(a_(22) - a_(21)a_(12)/a_(11)) & 1 \end{bmatrix} \cdot e_(31) \cdot e_(21) \cdot A\]`

After applying these elimination matrices in sequence, matrix \(A\) is transformed into its upper triangular form.

The complete question is probably:
How can the elimination matrices \(e_{21}\), \(e_{31}\), and \(e_{32}\) be utilized to transform matrix \(A\) into its upper triangular form? Provide a detailed explanation of the process and the specific operations performed by each elimination matrix.