# Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$

## Problem 485

Let
$A=\begin{bmatrix} 1 & -14 & 4 \\ -1 &6 &-2 \\ -2 & 24 & -7 \end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix} 4 \\ -1 \\ -7 \end{bmatrix}.$ Find $A^{10}\mathbf{v}$.

You may use the following information without proving it.
The eigenvalues of $A$ are $-1, 0, 1$. The eigenspaces are given by
$E_{-1}=\Span\left\{\, \begin{bmatrix} 3 \\ -1 \\ -5 \end{bmatrix} \,\right\}, \quad E_{0}=\Span\left\{\, \begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix} \,\right\}, \quad E_{1}=\Span\left\{\, \begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix} \,\right\}.$

(The Ohio State University, Linear Algebra Final Exam Problem)

## Solution.

Since $A$ has three distinct eigenvalues, the eigenvectors
$\begin{bmatrix} 3 \\ -1 \\ -5 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix} , \begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix}$ form a basis of $\R^3$.
Thus, we can express the vector $\mathbf{v}$ as a linear combination
$\mathbf{v}=x\begin{bmatrix} 3 \\ -1 \\ -5 \end{bmatrix}+y\begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix}+z\begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix}$ for some scalars $x,y,z$.

We determine the values of $x, y, z$ by Gauss-Jordan elimination.
The augmented matrix of the system is
\begin{align*}
\left[\begin{array}{rrr|r}
3 & -2 & -4 & 4 \\
-1 &1 & 2 & -1 \\
-5 & 4 & 7 & -7
\end{array} \right].
\end{align*}
Applying the elementary row operations, we obtain
\begin{align*}
\left[\begin{array}{rrr|r}
3 & -2 & -4 & 4 \\
-1 &1 & 2 & -1 \\
-5 & 4 & 7 & -7
\end{array} \right] \xrightarrow[\text{Then } -R_1]{R_1\leftrightarrow R_2}
\left[\begin{array}{rrr|r}
1 & -1 & -2 & 1 \\
3 & -2 & -4 & 4 \\
-5 & 4 & 7 & -7
\end{array} \right]\6pt] \xrightarrow[R_3+5R_1]{R_2-3R_1} \left[\begin{array}{rrr|r} 1 & -1 & -2 & 1 \\ 0 & 1 & 2 & 1 \\ 0 & -1 & -3 & -2 \end{array} \right] \xrightarrow[\text{Then } -R_3]{\substack{R_1+R_2\\R_3+R_2}} \left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 1 \end{array} \right]\\[6pt] \xrightarrow{R_2-2R_3} \left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 1 \end{array} \right]. \end{align*} So we have \[x=2, y=-1, z=1, and the linear combination is
$\mathbf{v}=2\begin{bmatrix} 3 \\ -1 \\ -5 \end{bmatrix}-\begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix}+\begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix}.$

Recall that if $\lambda$ is an eigenvalue and $\mathbf{u}$ is a corresponding eigenvector of $A$, then we have
$A^{10}\mathbf{u}=\lambda^{10}\mathbf{u}.$ Using this property, we compute
\begin{align*}
A^{10}\mathbf{v}&=A^{10}\left(\,2\begin{bmatrix}
3 \\
-1 \\
-5
\end{bmatrix}-\begin{bmatrix}
-2 \\
1 \\
4
\end{bmatrix}+\begin{bmatrix}
-4 \\
2 \\
7
\end{bmatrix} \,\right)\6pt] &=2A^{10}\begin{bmatrix} 3 \\ -1 \\ -5 \end{bmatrix}-A^{10}\begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix}+A^{10}\begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix}\\[6pt] &=2(-1)^{10}\begin{bmatrix} 3 \\ -1 \\ -5 \end{bmatrix}-0^{10}\begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix}+(1)^{10}\begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix}\\[6pt] &=2\begin{bmatrix} 3 \\ -1 \\ -5 \end{bmatrix}+\begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix}\\[6pt] &=\begin{bmatrix} 2 \\ 0 \\ -3 \end{bmatrix}. \end{align*} In summary, we obtain \[A^{10}\mathbf{v}=\begin{bmatrix} 2 \\ 0 \\ -3 \end{bmatrix}.

## Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)

This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).

The other problems can be found from the links below.

### More from my site

• Given All Eigenvalues and Eigenspaces, Compute a Matrix Product Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces $E_2=\Span\left \{\quad \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix} 1 \\ 2 \\ 1 \\ 1 […] • Diagonalize a 2 by 2 Matrix if Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}$ is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
• Diagonalize the 3 by 3 Matrix Whose Entries are All One Diagonalize the matrix $A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &1 &1 \\ 1 & 1 & 1 \end{bmatrix}.$ Namely, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
• Eigenvalues of a Hermitian Matrix are Real Numbers Show that eigenvalues of a Hermitian matrix $A$ are real numbers. (The Ohio State University Linear Algebra Exam Problem)   We give two proofs. These two proofs are essentially the same. The second proof is a bit simpler and concise compared to the first one. […]
• 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 […]
• Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable For which values of constants $a, b$ and $c$ is the matrix $A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}$ diagonalizable? (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the […]
• Find All the Eigenvalues of 4 by 4 Matrix Find all the eigenvalues of the matrix $A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 &0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}.$ (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. We compute the […]
• Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue Let $A=\begin{bmatrix} 1 & 2 & 1 \\ -1 &4 &1 \\ 2 & -4 & 0 \end{bmatrix}.$ The matrix $A$ has an eigenvalue $2$. Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$. (The Ohio State University, Linear Algebra Final Exam […]

### 4 Responses

1. 06/28/2017

[…] Compute $A^{10}mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ […]

2. 10/11/2017

[…] Compute $A^{10}mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ […]

3. 10/16/2017

[…] Compute $A^{10}mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ […]

4. 11/20/2017

[…] Compute $A^{10}mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ […]

##### Given the Characteristic Polynomial, Find the Rank of the Matrix

Let $A$ be a square matrix and its characteristic polynomial is give by $p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).$ Find the rank of $A$. (The...

Close