First slide

Elementary row operations (Transformation)

Question

A value of $b$ for which the rank of the matrix is $A = [\begin{matrix} 1 & 1 & - 1 & 0 \\ 4 & 4 & - 3 & 1 \\ b & 2 & 2 & 2 \\ 9 & 9 & b & 3 \end{matrix}]$ is 3, is

Moderate

Solution

$[\begin{matrix} 1 & 1 & - 1 & 0 \\ 4 & 4 & - 3 & 1 \\ b & 2 & 2 & 2 \\ 9 & 9 & b & 3 \end{matrix}]$ $\begin{array}{l} R_{2} \to R_{2} - 4 R_{1} \\ R_{3} \to R_{3} - b R_{1} \\ R_{4} \to R_{4} - 9 R_{1} \end{array}$
$~ [\begin{matrix} 1 & 1 & - 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 2 - b & 2 + b & 2 \\ 0 & 0 & b + 9 & 3 \end{matrix}]$ $R_{2} \leftrightarrow R_{3}$
$~ [\begin{matrix} 1 & 1 & - 1 & 0 \\ 0 & 2 - b & 2 + b & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & b + 9 & 3 \end{matrix}]$ $R_{4} \to R_{4} - 3 R_{3}$
$~ [\begin{matrix} 1 & 1 & - 1 & 0 \\ 0 & 2 - b & 2 + b & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & b + 6 & 0 \end{matrix}]$
Rank of $A = 3 \Rightarrow b + 6 = 0$
$\Rightarrow b = - 6$

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