To see that $M$ is singular, we will row-reduce the augmented matrix:
\[\left[\begin{array}{rr|r} 1 & 4 & 0 \\ 3 & 12 & 0 \end{array} \right] \xrightarrow{R_2 – 3 R_1} \left[\begin{array}{rr|r} 1 & 4 & 0 \\ 0 & 0 & 0 \end{array} \right].\]
The row-reduced matrix has a row of all $0$’s, and hence is singular.

(b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$.

We see that the vector $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ must satisfy the equation $v_1 + 4 v_2 = 0$.

There are many solutions. One solution, for example, is found by setting $v_1 = 1$ and solve $v_2 = – \frac{1}{4}$.

If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent
Let $V$ be a subspace of $\R^n$.
Suppose that
\[S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}\]
is a spanning set for $V$.
Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.
We give two proofs. The essential ideas behind […]

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(b) Suppose that a $3\times 3$ homogeneous system of linear equations has a solution $x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system […]

Find Values of $h$ so that the Given Vectors are Linearly Independent
Find the value(s) of $h$ for which the following set of vectors
\[\left \{ \mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
h \\
1 \\
-h
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
1 \\
2h \\
3h+1
[…]

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Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent.
\begin{align*}
S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\},
\end{align*}
where
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
3 \\
1
\end{bmatrix}, […]

Find All the Values of $x$ so that a Given $3\times 3$ Matrix is Singular
Find all the values of $x$ so that the following matrix $A$ is a singular matrix.
\[A=\begin{bmatrix}
x & x^2 & 1 \\
2 &3 &1 \\
0 & -1 & 1
\end{bmatrix}.\]
Hint.
Use the fact that a matrix is singular if and only if its determinant is […]

If a Matrix is the Product of Two Matrices, is it Invertible?
(a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as
\[A=BC,\]
where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix.
Prove that the matrix $A$ cannot be invertible.
(b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be […]

Find All Values of $x$ so that a Matrix is Singular
Let
\[A=\begin{bmatrix}
1 & -x & 0 & 0 \\
0 &1 & -x & 0 \\
0 & 0 & 1 & -x \\
0 & 1 & 0 & -1
\end{bmatrix}\]
be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.
Hint.
Use the fact that a matrix is singular if and only […]