Find the inverse matrix of the $3\times 3$ matrix
\[A=\begin{bmatrix}
7 & 2 & -2 \\
-6 &-1 &2 \\
6 & 2 & -1
\end{bmatrix}\]
using the Cayley-Hamilton theorem.
Add to solve later 10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.
The quiz is designed to test your understanding of the basic properties of these topics.
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The solutions will be given after completing all the 10 problems.
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Add to solve later Consider the following system of linear equations
\begin{align*}
2x+3y+z&=-1\\
3x+3y+z&=1\\
2x+4y+z&=-2.
\end{align*}
(a) Find the coefficient matrix $A$ for this system.
(b) Find the inverse matrix of the coefficient matrix found in (a)
(c) Solve the system using the inverse matrix $A^{-1}$.
Add to solve later Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors
\[\begin{bmatrix}
1 \\
0
\end{bmatrix} \text{ and } \begin{bmatrix}
2 \\
1
\end{bmatrix},\]
respectively.
Add to solve later Find the inverse matrix of the matrix
\[A=\begin{bmatrix}
1 & 1 & 2 \\
9 &2 &0 \\
5 & 0 & 3
\end{bmatrix}\]
using the Cayley–Hamilton theorem.
Add to solve later Suppose $A$ is a positive definite symmetric $n\times n$ matrix.
(a) Prove that $A$ is invertible.
(b) Prove that $A^{-1}$ is symmetric.
(c) Prove that $A^{-1}$ is positive-definite.
(MIT, Linear Algebra Exam Problem)
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Add to solve later Determine the values of $x$ so that the matrix
\[A=\begin{bmatrix}
1 & 1 & x \\
1 &x &x \\
x & x & x
\end{bmatrix}\]
is invertible.
For those values of $x$, find the inverse matrix $A^{-1}$.
Add to solve later Let
\[A=\begin{bmatrix}
3 & -12 & 4 \\
-1 &0 &-2 \\
-1 & 5 & -1
\end{bmatrix}.\]
Then find all eigenvalues of $A^5$. If $A$ is invertible, then find all the eigenvalues of $A^{-1}$.
Add to solve later Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying
\[T\left(\, \begin{bmatrix}
1 \\
2
\end{bmatrix}\,\right)=\begin{bmatrix}
3 \\
4 \\
5
\end{bmatrix} \text{ and } T\left(\, \begin{bmatrix}
0 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}.\]
Find a general formula for
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \,\right).\]
(The Ohio State University, Linear Algebra Math 2568 Exam Problem)
Add to solve later Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a basis of the vector space $\R^2$, where
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
1
\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}
1 \\
-1
\end{bmatrix}.\]
The action of a linear transformation $T:\R^2\to \R^3$ on the basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ is given by
\begin{align*}
T(\mathbf{v}_1)=\begin{bmatrix}
2 \\
4 \\
6
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
0 \\
8 \\
10
\end{bmatrix}.
\end{align*}
Find the formula of $T(\mathbf{x})$, where
\[\mathbf{x}=\begin{bmatrix}
x \\
y
\end{bmatrix}\in \R^2.\]
Add to solve later Let $T$ be the linear transformation from the $3$-dimensional vector space $\R^3$ to $\R^3$ itself satisfying the following relations.
\begin{align*}
T\left(\, \begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix} \,\right)
=\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}, \qquad T\left(\, \begin{bmatrix}
2 \\
3 \\
5
\end{bmatrix} \, \right) =
\begin{bmatrix}
0 \\
2 \\
-1
\end{bmatrix}, \qquad
T \left( \, \begin{bmatrix}
0 \\
1 \\
2
\end{bmatrix} \, \right)=
\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}.
\end{align*}
Then for any vector
\[\mathbf{x}=\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}\in \R^3,\]
find the formula for $T(\mathbf{x})$.
Add to solve later Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix.
Prove the following statements.
(a) If $A$ is similar to $B$, then $B$ is similar to $A$.
(b) $A$ is similar to itself.
(c) If $A$ is similar to $B$ and $B$ is similar to $C$, then $A$ is similar to $C$.
(d) If $A$ is similar to the identity matrix $I$, then $A=I$.
(e) If $A$ or $B$ is nonsingular, then $AB$ is similar to $BA$.
(f) If $A$ is similar to $B$, then $A^k$ is similar to $B^k$ for any positive integer $k$.
Add to solve later Suppose that $A$ is a real $n\times n$ matrix.
(a) Is it true that $A$ must commute with its transpose?
(b) Suppose that the columns of $A$ (considered as vectors) form an orthonormal set.
Is it true that the rows of $A$ must also form an orthonormal set?
(University of California, Berkeley, Linear Algebra Qualifying Exam)
Add to solve later Let $A$ be the coefficient matrix of the system of linear equations
\begin{align*}
-x_1-2x_2&=1\\
2x_1+3x_2&=-1.
\end{align*}
(a) Solve the system by finding the inverse matrix $A^{-1}$.
(b) Let $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}$ be the solution of the system obtained in part (a).
Calculate and simplify
\[A^{2017}\mathbf{x}.\]
(The Ohio State University, Linear Algebra Midterm Exam Problem)
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Add to solve later Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 1 & 2 \\
0 &0 &1 \\
1 & 0 & 1
\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.
(The Ohio State University, Linear Algebra Midterm Exam Problem)
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Add to solve later Let $A, B, C$ be the following $3\times 3$ matrices.
\[A=\begin{bmatrix}
1 & 2 & 3 \\
4 &5 &6 \\
7 & 8 & 9
\end{bmatrix}, B=\begin{bmatrix}
1 & 0 & 1 \\
0 &3 &0 \\
1 & 0 & 5
\end{bmatrix}, C=\begin{bmatrix}
-1 & 0\ & 1 \\
0 &5 &6 \\
3 & 0 & 1
\end{bmatrix}.\]
Then compute and simplify the following expression.
\[(A^{\trans}-B)^{\trans}+C(B^{-1}C)^{-1}.\]
(The Ohio State University, Linear Algebra Midterm Exam Problem)
Read solution
Add to solve later (a) Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 0 & 1 \\
1 &0 &0 \\
2 & 1 & 1
\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.
(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\]
where
\[B=\begin{bmatrix}
4 & 1\\
2& 6
\end{bmatrix}.\]
Verify that the matrix $A$ you obtained is actually a nonsingular matrix.
(The Ohio State University, Linear Algebra Midterm Exam Problem)
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Add to solve later Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be $2$-dimensional vectors and let $A$ be a $2\times 2$ matrix.
(a) Show that if $\mathbf{v}_1, \mathbf{v}_2$ are linearly dependent vectors, then the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly dependent.
(b) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors, can we conclude that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent?
(c) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors and $A$ is nonsingular, then show that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent.
Add to solve later Determine whether there exists a nonsingular matrix $A$ if
\[A^2=AB+2A,\]
where $B$ is the following matrix.
If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.
(a) \[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
1 & 2 & -2
\end{bmatrix}\]
(b) \[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
2 & 1 & -4
\end{bmatrix}.\]
Add to solve later Let $A$ be the following $3\times 3$ upper triangular matrix.
\[A=\begin{bmatrix}
1 & x & y \\
0 &1 &z \\
0 & 0 & 1
\end{bmatrix},\]
where $x, y, z$ are some real numbers.
Determine whether the matrix $A$ is invertible or not. If it is invertible, then find the inverse matrix $A^{-1}$.
Add to solve later 
