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A matrix has Eigen values 1 & 4 with Eigen vectors $\begin{pmatrix} 1\\ -1 \end{pmatrix}$ and $\begin{pmatrix} 2\\ 1 \end{pmatrix}$ respectively. The matrix M is ?
in Linear Algebra
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AX = $\lambda$X

Now, let A = $\begin{bmatrix} a & b\\ c & d \end{bmatrix}$

 

When $\lambda$=1 :

$\begin{bmatrix} a & b\\ c & d \end{bmatrix}$ . $\begin{bmatrix} 1\\ -1 \end{bmatrix}$ = 1 * $\begin{bmatrix} 1\\ -1 \end{bmatrix}$

$\therefore$ a - b = 1 and c -d = -1

 

When $\lambda$=4 :

$\begin{bmatrix} a & b\\ c & d \end{bmatrix}$ . $\begin{bmatrix} 2\\ 1 \end{bmatrix}$ = 4 * $\begin{bmatrix} 2\\ 1 \end{bmatrix}$

$\therefore$ 2a + b = 8 and 2c + d = 4

 

Solving simultaneously,we get:

A = $\begin{bmatrix} 3 & 2\\ 1 & 2 \end{bmatrix}$
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