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

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

Question
Matrices
asked 2020-12-15
Determine the null space of each of the following matrices:
\(\begin{pmatrix}1 & 2 &-3&-1 \\ -2 & -4 & 6 &3 \end{pmatrix}\)

Answers (1)

2020-12-16
Step 1
Given:
\(\begin{pmatrix}1 & 2 &-3&-1 \\ -2 & -4 & 6 &3 \end{pmatrix}\)
Step 2
The first step is to find the reduced row echelon form of the matrix , Swap matrix rows : \(R_1 \leftrightarrow R_2\)
\(=\begin{pmatrix}-2 & -4 &6&3 \\ 1 & 2 & -3 &-1 \end{pmatrix}\)
Cancelleading coefficient in row \(R_2\) by performing \(R_2 \leftarrow R_2 + \frac{1}{2} \cdot R_1\)
\(=\begin{pmatrix}-2 & -4 &6&3 \\ 0 & 0 & 0 &\frac{1}{2} \end{pmatrix}\)
Multiply matrix row by constant : \(R_2 \leftarrow 2 \cdot R_2\)
\(=\begin{pmatrix}-2 & -4 &6&3 \\ 0 & 0 & 0 &1 \end{pmatrix}\)
Cancelleading coefficient in row \(R_1\) by performing \(R_1 \leftarrow R_1 - 3 \cdot R_2\)
\(=\begin{pmatrix}-2 & -4 &6&0 \\ 0 & 0 & 0 &1 \end{pmatrix}\)
Multiply matrix row by constant : \(R_1 \leftarrow -\frac{1}{2} \cdot R_1\)
\(=\begin{pmatrix}1 & 2 &-3&0 \\ 0 & 0 & 0 &1 \end{pmatrix}\)
Step 3
\(\begin{bmatrix}3 \\ 1 \\ 0 \\0 \end{bmatrix} , \begin{bmatrix}-2 \\ 0 \\1 \\0 \end{bmatrix}\)
0

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