Exercise 1.1.19

Question

"Elimination" subtracts a number $\ell_{i j}$ times row $j$ from row $i:$ a "row operation." Show how those steps can reduce the matrix $A$ in Example 4 to $R$ (except that this row echelon form $R$ has a row of zeros). The rank won't change !
$$
A=\left[\begin{array}{lll}
1 & 3 & 8 \
1 & 2 & 6 \
0 & 1 & 2
\end{array}ight] \quad ightarrow \quad ightarrow \quad R=\left[\begin{array}{lll}
1 & 0 & 2 \
0 & 1 & 2 \
0 & 0 & 0
\end{array}ight]=\operatorname{rref}(A)
$$

Neil Z. · Accepted Answer

\begin{align*}
A &= \begin{bmatrix}
1 & 3 & 8 \
1 & 2 & 6 \
0 & 1 & 2 \
\end{bmatrix}\[10pt]
&= \begin{bmatrix}
1 & 3 & 8 \
0 & -1 & -2 \
0 & 1 & 2 \
\end{bmatrix}\[10pt]
&= \begin{bmatrix}
1 & 3 & 8 \
0 & -1 & -2 \
0 & 0 & 0 \
\end{bmatrix}\[10pt]
&= \begin{bmatrix}
1 & 0 & 2 \
0 & -1 & -2 \
0 & 0 & 0 \
\end{bmatrix}\[10pt]
&= \begin{bmatrix}
1 & 0 & 2 \
0 & 1 & 2 \
0 & 0 & 0 \
\end{bmatrix}\[5pt]
&= \text{rref}(A)\
\end{align*}

Exercise 1.1.19

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Exercise 1.1.19

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