First slide

Elementary row operations (Transformation)

Question

If the matrix $A = [\begin{matrix} 1 & 2 & 3 & 0 \\ 2 & 4 & 3 & 2 \\ 3 & 2 & 1 & 3 \\ 6 & 8 & 7 & α \end{matrix}]$ is of rank $3$ Then $α =$

Moderate

Solution

$A = [\begin{matrix} 1 & 2 & 3 & 0 \\ 2 & 4 & 3 & 2 \\ 3 & 2 & 1 & 3 \\ 6 & 8 & 7 & α \end{matrix}]$
$R_{2} - 2 R_{1} R_{3} - 3 R_{1} R_{4} - 6 R_{1}$ $~ [\begin{matrix} 1 & 2 & 3 & 0 \\ 0 & 0 & - 3 & 2 \\ 0 & - 4 & - 8 & 3 \\ 0 & - 4 & - 11 & α \end{matrix}]$
$R_{2} \to R_{3} R_{4} - R_{3}$ $~ [\begin{matrix} 1 & 2 & 3 & 0 \\ 0 & - 4 & - 8 & 3 \\ 0 & 0 & - 3 & 2 \\ 0 & 0 & - 3 & α - 3 \end{matrix}]$
$R_{4} - R_{3}$ $~ [\begin{matrix} 1 & 2 & 3 & 0 \\ 0 & - 4 & - 8 & 3 \\ 0 & 0 & - 3 & 2 \\ 0 & 0 & 0 & α - 5 \end{matrix}]$
Rank $A = 3 \Rightarrow α - 5 = 0 \Rightarrow α = 5$

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