Let $A$ be a singular $n\times n$ matrix.
Let
\[\mathbf{e}_1=\begin{bmatrix}
1 \\
0 \\
\vdots \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1 \\
\vdots \\
0
\end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix}
0 \\
0 \\
\vdots \\
1
\end{bmatrix}\]
be unit vectors in $\R^n$.

Prove that at least one of the following matrix equations
\[A\mathbf{x}=\mathbf{e}_i\]
for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$.

Assume on the contrary that each matrix equation $A\mathbf{x}=\mathbf{e}_i$ has a solution.
Let $\mathbf{b}_i\in \R^n$ be a solution of $A\mathbf{x}=\mathbf{e}_i$ for each $i=1, \dots, n$.
That is, we have
\[A\mathbf{b}_i=\mathbf{e}_i.\]
Let $B=[\mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n]$ be the $n\times n$ matrix whose $i$-th column vector is $\mathbf{b}_i$.

Then we have
\begin{align*}
AB&=A[\mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n]\\[6pt]
&=[A\mathbf{b}_1, A\mathbf{b}_2, \dots, A\mathbf{b}_n]\\[6pt]
&=[\mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n]=I,
\end{align*}
where $I$ is the $n\times n$ identity matrix.

Since $I$ is the nonsingular matrix, the matrix $A$ must also be nonsingular.
However this contradicts the assumption that $A$ is singular.
It follows that at least one of the matrix equations $A\mathbf{x}=\mathbf{e}_i$ has no solution.

Determine Conditions on Scalars so that the Set of Vectors is Linearly Dependent
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}, […]

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Let $A$ be a $3 \times 3$ matrix.
Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have
\[A\mathbf{x}=\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}, A\mathbf{y}=\begin{bmatrix}
0 \\
1 \\
0
[…]

Perturbation of a Singular Matrix is Nonsingular
Suppose that $A$ is an $n\times n$ singular matrix.
Prove that for sufficiently small $\epsilon>0$, the matrix $A-\epsilon I$ is nonsingular, where $I$ is the $n \times n$ identity matrix.
Hint.
Consider the characteristic polynomial $p(t)$ of the matrix $A$.
Note […]

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A square matrix $A$ is called idempotent if $A^2=A$.
(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.
Prove that $P$ is an idempotent matrix.
(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be […]

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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 […]

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 […]

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 […]