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

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

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
$$\begin{bmatrix}3 \\ 1 \\ 0 \\0 \end{bmatrix} , \begin{bmatrix}-2 \\ 0 \\1 \\0 \end{bmatrix}$$

Relevant Questions

Write out the system of equations that corresponds to each of the following augmented matrices:
(a)$$\begin{pmatrix}3 & 2&|&8 \\1 & 5&|&7 \end{pmatrix}$$
(b)$$\begin{pmatrix}5 & -2&1&|&3 \\2 & 3&-4&|&0 \end{pmatrix}$$
(c)$$\begin{pmatrix}2 & 1&4&|&-1 \\4 & -2&3&|&4 \\5 & 2&6&|&-1 \end{pmatrix}$$
(d)$$\begin{pmatrix}4 & -3&1&2&|&4 \\3 & 1&-5&6&|&5 \\1 & 1&2&4&|&8\\5 & 1&3&-2&|&7 \end{pmatrix}$$
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$$\begin{bmatrix}2 & 1 \\3 & 2 \end{bmatrix}$$
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$$\begin{bmatrix}1 & 1&-1&2 \\2 & 2&-3&1\\-1&-1&0&-5 \end{bmatrix}$$
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Let $$A=\begin{pmatrix}2 &1 \\6 & 4 \end{pmatrix}$$
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b) Express A as a product of elementary matrices
$$A=\begin{bmatrix}4i & \sqrt{-1} \\\sqrt{2}e^{i\pi/4} & 5\sin(53.13) \\\sqrt{-9} & \sqrt{1} \end{bmatrix}$$ and $$B=\begin{bmatrix}\ln e^3& \log_{x}{x^2}& \sqrt{-1}\ln e^i \\ i \log_{y}{y^{2i}} & 2e^{i\pi} & \ln e^i \end{bmatrix}$$
$$A=\begin{bmatrix}2&1&1 \\-1&-1&4 \end{bmatrix} , B=\begin{bmatrix}0 & 2 \\-4 & 1\\2&-3 \end{bmatrix} , C=\begin{bmatrix}6 & -1 \\3 & 0\\-2&5 \end{bmatrix} , D=\begin{bmatrix}2 & -3&4 \\-3& 1&-2 \end{bmatrix}$$