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 […]

How to Calculate and Simplify a Matrix Polynomial
Let $T=\begin{bmatrix}
1 & 0 & 2 \\
0 &1 &1 \\
0 & 0 & 2
\end{bmatrix}$.
Calculate and simplify the expression
\[-T^3+4T^2+5T-2I,\]
where $I$ is the $3\times 3$ identity matrix.
(The Ohio State University exam)
Hint.
Use the […]

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

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 […]

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 […]

Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$
Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities.
What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix?
(The Ohio State University, Linear Algebra Final Exam […]

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
[…]

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[…] Find Values of $a , b , c$ such that the Given Matrix is Diagonalizable […]

[…] Find Values of $a , b , c$ such that the Given Matrix is Diagonalizable […]