Determine All Matrices Satisfying Some Conditions on Eigenvalues and Eigenvectors

Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra

Problem 423

Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors
\[\begin{bmatrix}
1 \\
0
\end{bmatrix} \text{ and } \begin{bmatrix}
2 \\
1
\end{bmatrix},\] respectively.

 
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Solution.

Suppose that $A$ is a $2\times 2$ matrix having eigenvalues $2$ and $-1$ with corresponding eigenvectors
\[\begin{bmatrix}
1 \\
0
\end{bmatrix} \text{ and } \begin{bmatrix}
2 \\
1
\end{bmatrix},\] respectively.
Then since $A$ has two distinct eigenvalues, the matrix $A$ is diagonalizable.
As we know eigenvectors, we can diagonalize $A$ by the matrix
\[S:=\begin{bmatrix}
1 & 2\\
0& 1
\end{bmatrix}.\] That is, we have
\[S^{-1}AS=\begin{bmatrix}
2 & 0\\
0& -1
\end{bmatrix}.\] The inverse matrix of $S$ is given by
\[S^{-1}=\begin{bmatrix}
1 & -2\\
0& 1
\end{bmatrix}.\] It follows that we have
\begin{align*}
A&=S\begin{bmatrix}
1 & -2\\
0& 1
\end{bmatrix}S^{-1}\\[6pt] &=\begin{bmatrix}
2 & 0\\
0& -1
\end{bmatrix}
\begin{bmatrix}
1 & -2\\
0& 1
\end{bmatrix}
\begin{bmatrix}
1 & -2\\
0& 1
\end{bmatrix}\\[6pt] &=\begin{bmatrix}
2 & -6\\
0& 1
\end{bmatrix}.
\end{align*}

Therefore, the only matrix satisfying the given conditions is
\[A=\begin{bmatrix}
2 & -6\\
0& 1
\end{bmatrix}.\]


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