# Determine the null space of each of the following matrices: begin{pmatrix}1 & 2 &-3&-1 -2 & -4 & 6 &3 end{pmatrix}

Falak Kinney 2020-12-15 Answered
Determine the null space of each of the following matrices:
$\left(\begin{array}{cccc}1& 2& -3& -1\\ -2& -4& 6& 3\end{array}\right)$
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SchepperJ
Step 1
Given:
$\left(\begin{array}{cccc}1& 2& -3& -1\\ -2& -4& 6& 3\end{array}\right)$
Step 2
The first step is to find the reduced row echelon form of the matrix , Swap matrix rows : ${R}_{1}↔{R}_{2}$
$=\left(\begin{array}{cccc}-2& -4& 6& 3\\ 1& 2& -3& -1\end{array}\right)$
Cancelleading coefficient in row ${R}_{2}$ by performing ${R}_{2}←{R}_{2}+\frac{1}{2}\cdot {R}_{1}$
$=\left(\begin{array}{cccc}-2& -4& 6& 3\\ 0& 0& 0& \frac{1}{2}\end{array}\right)$
Multiply matrix row by constant : ${R}_{2}←2\cdot {R}_{2}$
$=\left(\begin{array}{cccc}-2& -4& 6& 3\\ 0& 0& 0& 1\end{array}\right)$
Cancelleading coefficient in row ${R}_{1}$ by performing ${R}_{1}←{R}_{1}-3\cdot {R}_{2}$
$=\left(\begin{array}{cccc}-2& -4& 6& 0\\ 0& 0& 0& 1\end{array}\right)$
Multiply matrix row by constant : ${R}_{1}←-\frac{1}{2}\cdot {R}_{1}$
$=\left(\begin{array}{cccc}1& 2& -3& 0\\ 0& 0& 0& 1\end{array}\right)$
Step 3
$\left[\begin{array}{c}3\\ 1\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}-2\\ 0\\ 1\\ 0\end{array}\right]$
Jeffrey Jordon