# Determine the null space of each of the following matrices: begin{bmatrix}1 & 1&-1&2 2 & 2&-3&1-1&-1&0&-5 end{bmatrix}

Determine the null space of each of the following matrices:
$\left[\begin{array}{cccc}1& 1& -1& 2\\ 2& 2& -3& 1\\ -1& -1& 0& -5\end{array}\right]$
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Step 1
Solve the given system Ax=0 using Gauss-jordan reduction method.
Step 2
Obtain the augmented matrix as shown below:
$\left[\begin{array}{cccccc}1& 1& -1& 2& |& 0\\ 2& 2& -3& 1& |& 0\\ -1& -1& 0& -5& |& 0\end{array}\right]\sim \left[\begin{array}{cccccc}1& 1& -1& 2& |& 0\\ 0& 0& -1& -3& |& 0\\ -1& -1& 0& -5& |& 0\end{array}\right]\left({R}_{2}\to {R}_{2}-2{R}_{1}\right)$
$\sim \left[\begin{array}{cccccc}1& 1& -1& 2& |& 0\\ 0& 0& -1& -3& |& 0\\ 0& 0& -1& -3& |& 0\end{array}\right]\left({R}_{3}\to {R}_{3}+{R}_{1}\right)$
$\sim \left[\begin{array}{cccccc}1& 1& -1& 2& |& 0\\ 0& 0& -1& -3& |& 0\\ 0& 0& 0& 0& |& 0\end{array}\right]\left({R}_{3}\to {R}_{3}-{R}_{2}\right)$
$\sim \left[\begin{array}{cccccc}1& 1& 0& 5& |& 0\\ 0& 0& -1& -3& |& 0\\ 0& 0& 0& 0& |& 0\end{array}\right]\left({R}_{1}\to {R}_{1}-{R}_{2}\right)$
$\sim \left[\begin{array}{cccccc}1& 1& 0& 5& |& 0\\ 0& 0& 1& 3& |& 0\\ 0& 0& 0& 0& |& 0\end{array}\right]\left({R}_{2}\to -{R}_{2}\right)$
Step 3
Then,
$\left[\begin{array}{cccc}1& 1& 0& 5\\ 0& 0& 1& 3\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right]=0$
${x}_{1}+{x}_{2}+5{x}_{4}=0$
${x}_{3}+3{x}_{4}=0$
Here the free variables are ${x}_{2},{x}_{4}$ thus , on setting ${x}_{2}=\alpha ,{x}_{4}=\beta$ we have , ${x}_{1}=-\alpha -5\beta ,{x}_{3}=-3\beta$
Thus , the null space N(A) contain all the vectors of the form:
$\left(-\alpha -5\beta ,\alpha ,-3\beta ,\beta {\right)}^{T}=\alpha \left(-1.1.0.0{\right)}^{T}+\beta \left(-5,0,-3,1{\right)}^{T},\alpha ,\beta \in \mathbb{R}$
Jeffrey Jordon