Question

A matrix M has eigen values 1 and 4 with corresponding eigen vectors (1,-1)^T ans (2,1)^T respectively. Then what will be M?

Answer 1

Answer on Question#38460 - Math - Matrix

Initial data:

$M = \left( \begin{array}{ll}a & b\\ c & d \end{array} \right)$ – Matrix M is a square matrix of dimension 2 because it has 2 eigen values.

$\lambda_{1} = 1;\lambda_{1} = 4$

$$v _ {1} = \left( \begin{array}{c} 1 \\ - 1 \end{array} \right); v _ {1} = \left( \begin{array}{c} 2 \\ 1 \end{array} \right)$$

Solution:

From definition of eigen vectors we can write^

$$\left\{ \begin{array}{l} M v _ {1} = \lambda_ {1} v _ {1} \\ M v _ {2} = \lambda_ {2} v _ {2} \end{array} \right.$$

Rewrite this in the extended form:

$$\left\{ \begin{array}{l l} \left( \begin{array}{c c} a & b \\ c & d \end{array} \right) \left( \begin{array}{c} 1 \\ - 1 \end{array} \right) = 1 \left( \begin{array}{c} 1 \\ - 1 \end{array} \right) \\ \left( \begin{array}{c c} a & b \\ c & d \end{array} \right) \left( \begin{array}{c} 2 \\ 1 \end{array} \right) = 4 \left( \begin{array}{c} 2 \\ 1 \end{array} \right) \end{array} \right.$$

From this we have a system of 4 equations with 4 unknowns. We can easy find this unknowns with the substitution method:

$$\left\{ \begin{array}{l} a - b = 1 \\ c - d = - 1 \\ 2 a + b = 8 \\ 2 c + d = 4 \end{array} \right. \quad = > \quad \left\{ \begin{array}{c} a = 1 + b \\ c = - 1 + d \\ 2 + 2 b + b = 8 \\ - 2 + 2 d + d = 4 \end{array} \right.$$ $$\left\{ \begin{array}{l} 3 b = 6 \\ 3 d = 6 \end{array} \right. \quad = > \quad \left\{ \begin{array}{l} b = 2 \\ d = 2 \end{array} \right.$$

The solution of the system is:

$$\left\{ \begin{array}{l} a = 3 \\ c = 1 \\ b = 2 \\ d = 2 \end{array} \right.$$

So, our matrix $M$ is:

$$M = \left( \begin{array}{c c} 3 & 2 \\ 1 & 2 \end{array} \right)$$