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GO Classes DPP | GATE CS, DA | Linear Algebra | Eigen Vectors

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4 4 votes

The Eigen values of a $2 \times 2$ matrix $A$ are $1, -2$, and its Eigen vectors $x_1$ and $x_2$ respectively.

The Eigen values and Eigen vectors of the matrix $A^{2} - 3A + 4I$  (where $I$ is the identity matrix) will be:

  1.   $2,14$ and $ x_1,x_2$
     
  2.   $2,14$ and $x_1+x_2 , x_1- x_2$
     
  3.   $2,0$ and $x_1,x_2$
     
  4.   $2,0$ and $x_1+x_2, x_1- x_2$

2 Answers

1 1 vote
GIven $A^2-3A+1$,

Eigen value $1: 1-3+4=2$,

Eigen value $2= 4+6+4=14$

We know $AX=\lambda X$

$A^2X=A \lambda X= \lambda^2X$

$A^2X-3AX=\lambda^2X-3 \lambda X $

$A^2X-3AX+4IX=A =\lambda^2X-3 \lambda X+4IX $

$(A^2-3A+4I)X=(\lambda^2-3 \lambda +4I)X $

 Eigenvalue of the matrix $(A^2 - 3A +4I)$ is $(  λ^2 - 3λ + 4)$ and the eigenvector is same as $A$ which is $x_1,x_2$.
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