3. Determine the null space of the following matrix: $$ \left(\begin{array}{cccc}-1 & -3 & 1 & 0 \ 3 & 0 & 6 & 9\end{array}\right) $$

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To determine the null space of the matrix

$$
A = \begin{pmatrix} -1 & -3 & 1 & 0 \ 3 & 0 & 6 & 9 \end{pmatrix},
$$

we need to solve the equation $$ A \mathbf{x} = \mathbf{0} $$, where $$ \mathbf{x} $$ is a vector in $$ \mathbb{R}^4 $$. We can express $$ \mathbf{x} $$ as

$$
\mathbf{x} = \begin{pmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{pmatrix}.
$$

This leads to the system of equations:

$$
\begin{pmatrix} -1 & -3 & 1 & 0 \ 3 & 0 & 6 & 9 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix}.
$$

This gives us the following equations:

1. $$-x_1 - 3x_2 + x_3 = 0$$
2. $$3x_1 + 6x_3 + 9x_4 = 0$$

We can simplify these equations. From the first equation, we can express $$ x_1 $$ in terms of $$ x_2 $$ and $$ x_3 $$:

$$
x_1 = -3x_2 + x_3.
$$

Substituting $$ x_1 $$ into the second equation:

$$
3(-3x_2 + x_3) + 6x_3 + 9x_4 = 0.
$$

This simplifies to:

$$
-9x_2 + 3x_3 + 6x_3 + 9x_4 = 0,
$$
$$
-9x_2 + 9x_3 + 9x_4 = 0.
$$

Dividing the entire equation by 9 gives:

$$
-x_2 + x_3 + x_4 = 0.
$$

From this, we can express $$ x_2 $$ in terms of $$ x_3 $$ and $$ x_4 $$:

$$
x_2 = x_3 + x_4.
$$

Now we have:

1. $$ x_1 = -3x_2 + x_3 = -3(x_3 + x_4) + x_3 = -3x_3 - 3x_4 + x_3 = -2x_3 - 3x_4 $$
2. $$ x_2 = x_3 + x_4 $$
3. $$ x_3 = x_3 $$
4. $$ x_4 = x_4 $$

Let $$ x_3 = t $$ and $$ x_4 = s $$ for parameters $$ t $$ and $$ s $$. Then we can express the solution vector $$ \mathbf{x} $$ as:

$$
\mathbf{x} = \begin{pmatrix} -2t - 3s \ t + s \ t \ s \end{pmatrix} = t \begin{pmatrix} -2 \ 1 \ 1 \ 0 \end{pmatrix} + s \begin{pmatrix} -3 \ 1 \ 0 \ 1 \end{pmatrix}.
$$

Thus, the null space of the matrix $$ A $$ is spanned by the vectors

$$
\begin{pmatrix} -2 \ 1 \ 1 \ 0 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} -3 \ 1 \ 0 \ 1 \end{pmatrix}.
$$

In conclusion, the null space of the matrix $$ A $$ is given by:

$$
\text{Null}(A) = \text{span} \left\{ \begin{pmatrix} -2 \ 1 \ 1 \ 0 \end{pmatrix}, \begin{pmatrix} -3 \ 1 \ 0 \ 1 \end{pmatrix} \right\}.
$$
The null space of the matrix is spanned by the vectors $$\begin{pmatrix} -2 \ 1 \ 1 \ 0 \end{pmatrix}$$ and $$\begin{pmatrix} -3 \ 1 \ 0 \ 1 \end{pmatrix}$$.

Determine the null space of the following matrix:

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