A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable
Problem 213
Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.
Add to solve laterLet $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.
Add to solve later Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix.
If
\[P^{-1}AP=\begin{bmatrix}
1 & 2 & 3 \\
0 &4 &5 \\
0 & 0 & 6
\end{bmatrix},\]
then find all the eigenvalues of the matrix $A^2$.
Read solution
Suppose the following information is known about a $3\times 3$ matrix $A$.
\[A\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix}=6\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix},
\quad
A\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}=3\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}, \quad
A\begin{bmatrix}
2 \\
-1 \\
0
\end{bmatrix}=3\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}.\]
(a) Find the eigenvalues of $A$.
(b) Find the corresponding eigenspaces.
(c) In each of the following questions, you must give a correct reason (based on the theory of eigenvalues and eigenvectors) to get full credit.
Is $A$ a diagonalizable matrix?
Is $A$ an invertible matrix?
Is $A$ an idempotent matrix?
(Johns Hopkins University Linear Algebra Exam)
Read solution
Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix.
Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity matrix.
Add to solve laterLet $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number.
Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, the $n$-dimensional vector space over $\F_p$. Therefore, $G_n$ acts on $\F_p^n$.
Let $e_n \in \F_p^n$ be the vector $(1,0, \dots,0)$.
(The so-called first standard basis vector in $\F_p^n$.)
Find the size of the $G_n$-orbit of $e_n$, and show that $\Stab_{G_n}(e_n)$ has order $|G_{n-1}|\cdot p^{n-1}$.
Conclude by induction that
\[|G_n|=p^{n^2}\prod_{i=1}^{n} \left(1-\frac{1}{p^i} \right).\]
For which choice(s) of the constant $k$ is the following matrix invertible?
\[A=\begin{bmatrix}
1 & 1 & 1 \\
1 &2 &k \\
1 & 4 & k^2
\end{bmatrix}.\]
(Johns Hopkins University, Linear Algebra Exam)
Read solution
A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.
(a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent.
(b) Let $P$ be an invertible $n \times n$ matrix and let $N$ be a nilpotent $n\times n$ matrix. Is the product $PN$ nilpotent? If so, prove it. If not, give a counterexample.
Add to solve later
Let A be the matrix
\[\begin{bmatrix}
1 & -1 & 0 \\
0 &1 &-1 \\
0 & 0 & 1
\end{bmatrix}.\]
Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse.
(The Ohio State University Linear Algebra Exam)
Add to solve laterSuppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$.
(a) If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not.
(b) Is $3\mathbf{v}$ an eigenvector of $A$? If so, what is the corresponding eigenvalue? If not, explain why not.
(Stanford University, Linear Algebra Exam)
Add to solve laterConsider the matrix
\[A=\begin{bmatrix}
1 & 2 & 1 \\
2 &5 &4 \\
1 & 1 & 0
\end{bmatrix}.\]
(a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why.
(b) Are the vectors
\[ \mathbf{A}_1=\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix}, \mathbf{A}_2=\begin{bmatrix}
2 \\
5 \\
1
\end{bmatrix},
\text{ and } \mathbf{A}_3=\begin{bmatrix}
1 \\
4 \\
0
\end{bmatrix}\]
linearly independent?
(c) Write the vector $\mathbf{b}=\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}$ as a linear combination of $\mathbf{A}_1$, $\mathbf{A}_2$, and $\mathbf{A}_3$.
(The Ohio State University, Linear Algebra Exam)
Add to solve laterLet $A$ and $B$ be $n \times n$ matrices.
Prove that the characteristic polynomials for the matrices $AB$ and $BA$ are the same.
Add to solve laterLet $A$ and $B$ are $n \times n$ matrices with real entries.
Assume that $A+B$ is invertible. Then show that
\[A(A+B)^{-1}B=B(A+B)^{-1}A.\]
(University of California, Berkeley Qualifying Exam)
Add to solve laterLet $A$ be an $m \times n$ real matrix.
Then the kernel of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$.
The kernel is also called the null space of $A$.
Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is invertible.
(Stanford University Linear Algebra Exam)
Add to solve laterLet
\[ A=\begin{bmatrix}
2 & 0 & 10 \\
0 &7+x &-3 \\
0 & 4 & x
\end{bmatrix}.\]
Find all values of $x$ such that $A$ is invertible.
(Stanford University Linear Algebra Exam)
Add to solve laterIn this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility.
That is, we will prove that:
(a) Show that if $A$ is invertible, then $A$ is nonsingular.
(b) Let $A, B, C$ be $n\times n$ matrices such that $AB=C$.
Prove that if either $A$ or $B$ is singular, then so is $C$.
(c) Show that if $A$ is nonsingular, then $A$ is invertible.
Add to solve laterAn $n \times n$ matrix $A$ is called nonsingular if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$.
Otherwise $A$ is called singular.
(a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.
(b) Show that if $A$ is nonsingular, then the column vectors of $A$ are linearly independent.
(c) Show that an $n \times n$ matrix $A$ is nonsingular if and only if the equation $A\mathbf{x}=\mathbf{b}$ has a unique solution for any vector $\mathbf{b}\in \R^n$.
Restriction
Do not use the fact that a matrix is nonsingular if and only if the matrix is invertible.
A square matrix $A$ is called idempotent if $A^2=A$.
Show that a square invertible idempotent matrix is the identity matrix.
Add to solve later