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

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

2021-03-07
Step 1
Consider the given matrix:
$$\begin{pmatrix}1 & 3 &-4 \\ 2 & -1 & -1 \\ -1 & -3 &4 \end{pmatrix}$$
Now solve the system of equations below to find the null spaces:
$$\begin{pmatrix}1 & 3 &-4 \\ 2 & -1 & -1 \\ -1 & -3 &4 \end{pmatrix}\begin{pmatrix}x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix}$$
Solve the above matrix using Gauss Elimination method . Find the row echelon form: $$R_2=R_2-2R_1$$
$$\begin{pmatrix}1 & 3 &-4 \\ 0& -7 & 7\\ -1 & -3 &4 \end{pmatrix}\begin{pmatrix}x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix}$$
$$R_3=R_3+R_1$$
$$\begin{pmatrix}1 & 3 &-4 \\ 0& -7 & 7\\ 0 & 0 &0 \end{pmatrix}\begin{pmatrix}x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix}$$
Step 2
we get equations:
$$-7x_2+7x_3=0$$
$$7x_2=7x_3$$
$$x_2=x_3$$ And $$x_1+3x_2-4x_3=0$$
$$x_1+3x_2-4x_2=0$$
$$x_1=x_2$$
Therefore null space is given by:
$$\begin{pmatrix}x_2 \\ x_2 \\ x_2 \end{pmatrix}=x_2\begin{pmatrix}1 \\ 1 \\ 1 \end{pmatrix}$$

### Relevant Questions

Determine the null space of each of the following matrices:
$$\begin{pmatrix}1 & 2 &-3&-1 \\ -2 & -4 & 6 &3 \end{pmatrix}$$
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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$$\begin{bmatrix}2 & 1 \\3 & 2 \end{bmatrix}$$
Determine the null space of each of the following matrices:
$$\begin{bmatrix}1 & 1&-1&2 \\2 & 2&-3&1\\-1&-1&0&-5 \end{bmatrix}$$

Let M be the vector space of $$2 \times 2$$ real-valued matrices.
$$M=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$
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$$B=\begin{pmatrix}-1 & -2\\1 & 2\\3&4 \end{pmatrix}$$
$$A=\begin{pmatrix}4 & 2&-2 \\2 & 2&-3\\-2&-3&14 \end{pmatrix} , b=\begin{pmatrix}10 , 5 , 4 \end{pmatrix}^T$$
Let $$A=\begin{pmatrix}2 &1 \\6 & 4 \end{pmatrix}$$
a) Express $$A^{-1}$$ as a product of elementary matrices