Compute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector
Let
\[A=\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix}
1\\
0
\end{bmatrix}.\]
Compute $A^{2017}\mathbf{u}$.
(The Ohio State University, Linear Algebra Exam)
Solution.
We first compute $A\mathbf{u}$. We […]

A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues
Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers.
Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$.
Hint.
Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix.
[…]

Show that the Given 2 by 2 Matrix is Singular
Consider the matrix $M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$.
(a) Show that $M$ is singular.
(b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$, where $\mathbf{0}$ is the $2$-dimensional zero vector.
Solution.
(a) Show […]

Find the Rank of the Matrix $A+I$ if Eigenvalues of $A$ are $1, 2, 3, 4, 5$
Let $A$ 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, […]

Describe the Range of the Matrix Using the Definition of the Range
Using the definition of the range of a matrix, describe the range of the matrix
\[A=\begin{bmatrix}
2 & 4 & 1 & -5 \\
1 &2 & 1 & -2 \\
1 & 2 & 0 & -3
\end{bmatrix}.\]
Solution.
By definition, the range $\calR(A)$ of the matrix $A$ is given […]

Trace of the Inverse Matrix of a Finite Order Matrix
Let $A$ be an $n\times n$ matrix such that $A^k=I_n$, where $k\in \N$ and $I_n$ is the $n \times n$ identity matrix.
Show that the trace of $(A^{-1})^{\trans}$ is the conjugate of the trace of $A$. That is, show that […]