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

Ohio State University exam problems and solutions in mathematics

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)

 
LoadingAdd to solve later

Sponsored Links


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.

  1. Find All the Eigenvalues of 4 by 4 Matrix
  2. Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue
  3. Diagonalize a 2 by 2 Matrix if Diagonalizable
  4. Find an Orthonormal Basis of the Range of a Linear Transformation
  5. The Product of Two Nonsingular Matrices is Nonsingular
  6. Determine Whether Given Subsets in ℝ4 R 4 are Subspaces or Not
  7. Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
  8. Find Values of $a , b , c$ such that the Given Matrix is Diagonalizable
  9. Idempotent Matrix and its Eigenvalues
  10. Diagonalize the 3 by 3 Matrix Whose Entries are All One
  11. Given the Characteristic Polynomial, Find the Rank of the Matrix
  12. Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ (This page)
  13. Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

LoadingAdd to solve later

Sponsored Links

More from my site

  • Given All Eigenvalues and Eigenspaces, Compute a Matrix ProductGiven 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 DiagonalizableDiagonalize 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 OneDiagonalize 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 NumbersEigenvalues 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 $A-aI$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 DiagonalizableFind 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 MatrixFind 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 EigenvalueFind 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 […]

You may also like...

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 $A$ […]

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Ohio State University exam problems and solutions in mathematics
Given the Characteristic Polynomial, Find the Rank of the Matrix

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

Close