Question

$A=\left[\begin{array}{ccc}2& 2& 1\\ 4& 5& 6\\ 6& 8& 9\end{array}\right]$, $B=\left[\begin{array}{ccc}2& 2& 1\\ 0& 1& 4\\ 0& 2& 6\end{array}\right]$

To convert matrix A into matrix B, the order of row operations are

• A. $R_1\to R_2-2R_1$, $R_3\to R_3-R_1$
• B. $R_2\to R_2-2R_1$, $R_3\to R_3-3R_1$
• C. $R_1\to R_1-2R_2$, $R_3\to R_1-R_3$
• D. $R_2\to R_3-2R_3$, $R_3\to R_1-R_1$

Answer 1

The correct option is B $R_2\to R_2-2R_1$, $R_3\to R_3-3R_1$

$A=\left[\begin{array}{ccc}2& 2& 1\\ 4& 5& 6\\ 6& 8& 9\end{array}\right]B=\left[\begin{array}{ccc}2& 2& 1\\ 0& 1& 4\\ 0& 2& 6\end{array}\right]$

Clearly in order to convert matrix A to B. Both $R_2$ and $R_3$ are changed w.r.t. $R_1$

$R_2^1\leftrightarrow X{R}_2-Y{R}_1$

For $R_2=4$ and $R_1=2$ then $R_2^1=0$

$\Rightarrow 0=X4-2Y$

$\Rightarrow 2X=Y$

For $R_2=5$ and $R_1=2$ then $R_2^1=1$

$1=X\left(5\right)-Y\left(2\right)$

$\Rightarrow X=1Y=2$

Relation is $R_2\leftrightarrow R_2-2R_2$

Parallely $R_3\to R_3-3R_1$