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

Determine the null space of each of the following matrices:
$\left(\begin{array}{ccc}1& 3& -4\\ 2& -1& -1\\ -1& -3& 4\end{array}\right)$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

SoosteethicU
Step 1
Consider the given matrix:
$\left(\begin{array}{ccc}1& 3& -4\\ 2& -1& -1\\ -1& -3& 4\end{array}\right)$
Now solve the system of equations below to find the null spaces:
$\left(\begin{array}{ccc}1& 3& -4\\ 2& -1& -1\\ -1& -3& 4\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$
Solve the above matrix using Gauss Elimination method . Find the row echelon form: ${R}_{2}={R}_{2}-2{R}_{1}$
$\left(\begin{array}{ccc}1& 3& -4\\ 0& -7& 7\\ -1& -3& 4\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$
${R}_{3}={R}_{3}+{R}_{1}$
$\left(\begin{array}{ccc}1& 3& -4\\ 0& -7& 7\\ 0& 0& 0\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$
Step 2
we get equations:
$-7{x}_{2}+7{x}_{3}=0$
$7{x}_{2}=7{x}_{3}$
${x}_{2}={x}_{3}$ And ${x}_{1}+3{x}_{2}-4{x}_{3}=0$
${x}_{1}+3{x}_{2}-4{x}_{2}=0$
${x}_{1}={x}_{2}$
Therefore null space is given by:
$\left(\begin{array}{c}{x}_{2}\\ {x}_{2}\\ {x}_{2}\end{array}\right)={x}_{2}\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$
Jeffrey Jordon