The Matrix $[A_1, \dots, A_{n-1}, A\mathbf{b}]$ is Always Singular, Where $A=[A_1,\dots, A_{n-1}]$ and $\mathbf{b}\in \R^{n-1}$. Problem 560

Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector.
Then the product $A\mathbf{b}$ is an $n$-dimensional vector.
Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$.

Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$. Add to solve later

Definition/Hint.

An $n\times n$ matrix $B$ is nonsingular if $A\mathbf{x}=\mathbf{0}, \mathbf{x}\in \R^n$ implies that $\mathbf{x}=\mathbf{0}$.
Otherwise, the matrix $B$ is called singular.

Namely, the matrix $B$ is singular if there exists a nonzero vector $\mathbf{v}$ such that $B\mathbf{v}=\mathbf{0}$.

You may use the fact that a matrix is nonsingular if and only if its column vectors are lienarly independent.

We give two proofs. The first one uses the definition of a singular matrix. The second one uses the fact mentioned above.

Proof (Using Defition).

Let
$\mathbf{b}=\begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_{n-1} \end{bmatrix}.$ Then we have
$A\mathbf{b}=b_1A_1+b_2A_2+\cdots+b_{n-1}A_{n-1} \tag{*}$ using the column vectors $A_i$ of $A$.

Let us define the $n$-dimensional vector $\mathbf{v}$ to be
$\mathbf{v}=\begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_{n-1}\\ -1 \end{bmatrix}.$

Since the last entry of $\mathbf{v}$ is $-1$, the vector $\mathbf{v}$ is nonzero.
We calculate
\begin{align*}
B\mathbf{v}&=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}] \begin{bmatrix}
b_1 \\
b_2 \\
\vdots \\
b_{n-1}\\
-1
\end{bmatrix}\6pt] &=b_1A_1+b_2A_2+\cdots+b_{n-1}A_{n-1}+(-1)A\mathbf{b}\\[6pt] &\stackrel{(*)}{=} b_1A_1+b_2A_2+\cdots+b_{n-1}A_{n-1}-(b_1A_1+b_2A_2+\cdots+b_{n-1}A_{n-1})\\ &=\mathbf{0}, \end{align*} where \mathbf{0} is the n-dimensional zero vector. Since we have B\mathbf{v}=\mathbf{0} for a nonzero vector \mathbf{v}, we conclude that the matrix B is singular. Proof (Using the Fact). Note that a square matrix is singular if and only if its columns vectors are linearly dependent. We prove that the column vectors A_1, A_2, \cdots, A_{n-1}, A\mathbf{b} of the matrix B are linearly dependent. Let \[\mathbf{b}=\begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_{n-1} \end{bmatrix}. Then we have
$A\mathbf{b}=b_1A_1+b_2A_2+\cdots+b_{n-1}A_{n-1}.$

Thus, we have the linear combination of column vectors of $B$
$b_1A_1+b_2A_2+\cdots+b_{n-1}A_{n-1}-A\mathbf{b}=\mathbf{0}.$ Since the coefficient in front of the vector $A\mathbf{b}$ is $-1$, the left hand side is a nontrivial linear combination of column vectors of $B$.
This implies that the column vectors of $B$ are linearly dependent, hence the matrix $B$ is singular. Add to solve later

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