Let $B$ be the $3\times 3$ matrix whose columns are the vectors $\mathbf{x},\mathbf{y}, \mathbf{z}$, that is,
\[B=[\mathbf{x} \mathbf{y} \mathbf{z}].\]
Then we have
\[AB=\begin{bmatrix}
1 & 0 & 1 \\
0 &1 &1 \\
1 & 0 & 1
\end{bmatrix}.\]
Then we have
\[\det(A)\det(B)=\det(AB)=\begin{vmatrix}
1 & 0 & 1 \\
0 &1 &1 \\
1 & 0 & 1
\end{vmatrix}=0.\]
(If two rows are equal, then the determinant is zero. Or you may compute the determinant by the second column cofactor expansion.)
Note that the column vectors of $B$ are linearly independent, and hence $B$ is nonsingular matrix. Thus the $\det(B)\neq 0$.
Therefore the determinant of $A$ must be zero.
we have
\[A\mathbf{x}+A\mathbf{y}=A\mathbf{z}.\]
It follows that we have
\[A(\mathbf{x}+\mathbf{y}-\mathbf{z})=\mathbf{0}.\]
Since the vectors $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent, the linear combination $\mathbf{x}+\mathbf{y}-\mathbf{z} \neq \mathbf{0}$.
Hence the matrix $A$ is singular, and the determinant of $A$ is zero.
(Recall that a matrix $A$ is singular if and only if there exist nonzero vector $\mathbf{v}$ such that $A\mathbf{u}=\mathbf{0}$.)
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\[\left \{ \mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
h \\
1 \\
-h
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
1 \\
2h \\
3h+1
[…]
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1 & -x & 0 & 0 \\
0 &1 & -x & 0 \\
0 & 0 & 1 & -x \\
0 & 1 & 0 & -1
\end{bmatrix}\]
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Hint.
Use the fact that a matrix is singular if and only […]
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(The Ohio State University, Linear Algebra Final Exam […]
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\begin{align*}
S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\},
\end{align*}
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\[\mathbf{v}_1=\begin{bmatrix}
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3 \\
1
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For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
\[A=\begin{bmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta &\cos\theta &0 \\
0 & 0 & 1
\end{bmatrix}.\]
(a) Find the determinant of the matrix $A$.
(b) Show that $A$ is an […]
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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.
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Maximize the Dimension of the Null Space of $A-aI$
Let
\[ A=\begin{bmatrix}
5 & 2 & -1 \\
2 &2 &2 \\
-1 & 2 & 5
\end{bmatrix}.\]
Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.
Your score of this problem is equal to that […]
Let \[A=\begin{bmatrix} 1 & -1\\ 2& 3 \end{bmatrix}.\] Find the eigenvalues and the eigenvectors of the matrix \[B=A^4-3A^3+3A^2-2A+8E.\] (Nagoya University...