Check the solution to "Construct a 3×3 matrix, not in echelon form,..."

Ayaana Buck

Answered question

2021-10-20

Construct a

3 \times 3

matrix, not in echelon form, whose columns span

R^{3}

. Show that the matrix you construct has the desired property.

Answer & Explanation

wheezym

Skilled2021-10-21Added 103 answers

Step 1
Let
$A = [\begin{matrix} 2 & 1 & 1 \\ 2 & 2 & 0 \\ 2 & 0 & 3 \end{matrix}]$
be matrix is not in echelon form
Reduce matrix to reduced echelon form:
$[\begin{matrix} 2 & 1 & 1 \\ 2 & 2 & 0 \\ 2 & 0 & 3 \end{matrix}]$
, multiply first row with -1 and add to second and third
$\sim [\begin{matrix} 2 & 1 & 1 \\ 0 & 1 & - 1 \\ 0 & - 1 & 2 \end{matrix}]$
, add second row to third
$\sim [\begin{matrix} 2 & 1 & 1 \\ 0 & 1 & - 1 \\ 0 & 0 & 1 \end{matrix}]$
, divide first row with 2
$\sim [\begin{matrix} 1 & \frac{1}{2} & \frac{1}{2} \\ 0 & 1 & - 1 \\ 0 & 0 & 1 \end{matrix}]$
add third row to second and multiply third row with $- \frac{1}{2}$ and add lo first
$\sim [\begin{matrix} 1 & \frac{1}{2} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$
, multiply second row with $- \frac{1}{2}$ and add to first
$\sim [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$
Since there are 3 pivot columns they span $R^{3}$
Answer
$A = [\begin{matrix} 2 & 1 & 1 \\ 2 & 2 & 0 \\ 2 & 0 & 3 \end{matrix}]$

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