User row operations to change the matrix to reduced form.
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Question

Quizplus · Accepted Answer

We need to use row operations to transform the matrix to reduced form. Here are the steps:

$ \begin{pmatrix}1 & 1 & 2 \ 2 & 2 & 5 \ 3 & 3 & 8 \end{pmatrix} $

R2 - 2R1 $ightarrow$ R2

$ \begin{pmatrix}1 & 1 & 2 \ 1 & 0 & 1 \ 3 & 3 & 8 \end{pmatrix} $

R3 - 3R1 $ightarrow$ R3

$ \begin{pmatrix}1 & 1 & 2 \ 1 & 0 & 1 \ 0 & 0 & 2 \end{pmatrix} $

R1 - R2 $ightarrow$ R1

$ \begin{pmatrix}0 & 1 & 1 \ 1 & 0 & 1 \ 0 & 0 & 2 \end{pmatrix} $

R2 $\leftrightarrow$ R3

$ \begin{pmatrix}0 & 1 & 1 \ 0 & 0 & 2 \ 1 & 0 & 1 \end{pmatrix} $

R1 $\leftrightarrow$ R2

$ \begin{pmatrix}0 & 0 & 2 \ 0 & 1 & 1 \ 1 & 0 & 1 \end{pmatrix} $

Now the matrix is in reduced form. The only choice that matches this reduced form is C). By performing row operations, we can reduce the matrix to row echelon form first, which is
$$\begin{pmatrix}1 & -1 & 1 & 1 \ 0 & 2 & 0 & 0 \ 0 & 0 & 1 & 1 \ 0 & 0 & 0 & 0\end{pmatrix}.$$ Then, we can further reduce it to reduced row echelon form by using row operations again:
$$\begin{pmatrix}1 & 0 & 1 & 1 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 1 \ 0 & 0 & 0 & 0\end{pmatrix}.$$This matches choice C.

User Row Operations to Change the Matrix to Reduced Form