Calculate $A^{10}$ for a Given Matrix $A$
Problem 41
Find $A^{10}$, where $A=\begin{bmatrix}
4 & 3 & 0 & 0 \\
3 &-4 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1
\end{bmatrix}$.
(Harvard University Exam)
Add to solve laterFind $A^{10}$, where $A=\begin{bmatrix}
4 & 3 & 0 & 0 \\
3 &-4 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1
\end{bmatrix}$.
(Harvard University Exam)
Add to solve laterFind a basis for the subspace $W$ of all vectors in $\R^4$ which are perpendicular to the columns of the matrix
\[A=\begin{bmatrix}
11 & 12 & 13 & 14 \\
21 &22 & 23 & 24 \\
31 & 32 & 33 & 34 \\
41 & 42 & 43 & 44
\end{bmatrix}.\]
(Harvard University 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 laterSuppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues.
Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix.
(Stanford University Linear Algebra Exam)
See below for a generalized problem.
Add to solve laterLet $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix.
(UCB-University of California, Berkeley, Exam)
Add to solve later(a) Let
\[A=\begin{bmatrix}
a_{11} & a_{12}\\
a_{21}& a_{22}
\end{bmatrix}\]
be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum of the entries in each row is $1$.
(Such a matrix is called (right) stochastic matrix (also termed probability matrix, transition matrix, substitution matrix, or Markov matrix).)
Then prove that the matrix $A$ has an eigenvalue $1$.
(b) Find all the eigenvalues of the matrix
\[B=\begin{bmatrix}
0.3 & 0.7\\
0.6& 0.4
\end{bmatrix}.\]
(c) For each eigenvalue of $B$, find the corresponding eigenvectors.
Add to solve laterSuppose that $S$ is a fixed invertible $3$ by $3$ matrix. This question is about all the matrices $A$ that are diagonalized by $S$, so that $S^{-1}AS$ is diagonal. Show that these matrices $A$ form a subspace of $3$ by $3$ matrix space.
(MIT-Massachusetts Institute of Technology 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.
Find all eigenvalues of the following $n \times n$ matrix.
\[
A=\begin{bmatrix}
0 & 0 & \cdots & 0 &1 \\
1 & 0 & \cdots & 0 & 0\\
0 & 1 & \cdots & 0 &0\\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0&\cdots & 1& 0 \\
\end{bmatrix}
\]
Let $A$ be an $n \times n$ matrix such that $\tr(A^n)=0$ for all $n \in \N$.
Then prove that $A$ is a nilpotent matrix. Namely there exist a positive integer $m$ such that $A^m$ is the zero matrix.
Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix.
(a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of $B$.
(b) Show that $\tr(AA^{\trans})$ is the sum of the square of the entries of $A$.
(c) Show that if $A$ is nonzero symmetric matrix, then $\tr(A^2)>0$.
Add to solve laterLet $A$ and $B$ be $n\times n$ matrices.
Suppose that these matrices have a common eigenvector $\mathbf{x}$.
Show that $\det(AB-BA)=0$.
Read solution
Let $A$ be an $n \times n$ real matrix. Prove the followings.
(a) The matrix $AA^{\trans}$ is a symmetric matrix.
(b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.
(c) The matrix $AA^{\trans}$ is non-negative definite.
(An $n\times n$ matrix $B$ is called non-negative definite if for any $n$ dimensional vector $\mathbf{x}$, we have $\mathbf{x}^{\trans}B \mathbf{x} \geq 0$.)
(d) All the eigenvalues of $AA^{\trans}$ is non-negative.
Add to solve laterLet $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.
Show that
(1) $$\det(A)=\prod_{i=1}^n \lambda_i$$
(2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$
Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix $A$.
Namely, prove that (1) the determinant of $A$ is the product of its eigenvalues, and (2) the trace of $A$ is the sum of the eigenvalues.
Read solution
Let $A= \begin{bmatrix}
1 & 2\\
2& 1
\end{bmatrix}$.
Compute $A^n$ for any $n \in \N$.
Let $A=\begin{bmatrix}
a & 0\\
0& b
\end{bmatrix}$.
Show that
(1) $A^n=\begin{bmatrix}
a^n & 0\\
0& b^n
\end{bmatrix}$ for any $n \in \N$.
(2) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix.
Show that $B^n=S^{-1}A^n S$ for any $n \in \N$
Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.
Add to solve later