Linear algebra

Question

Given that the matrix $A=\begin{bmatrix} a_{11} &a_{12} \\ a_{21} &a_{22} \end{bmatrix}$ and Eigen value are $1,-2$ and Corresponding Eigen Vectors are $X_{1}=\begin{bmatrix} 1\\2 \end{bmatrix}$ and  $X_{2}=\begin{bmatrix} 9\\1 \end{bmatrix}$

Find the $1)\sum a_{ij},$when $i=j$

             $2)Det(A)$

             $3)$Sum of all Elements in the given matrix

Comments

I have Problem on $3)$??

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is it

3) 27/17

a11= 1/17, a12= 8/17, a21= 2/17, a22=16/17

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$3)$No, the answer is $\frac{4}{17}$

How you got, can you share your method?

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ok is it for 1) is 1 ?

simply AX=eigen value*X then solve equation

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for $1)$ answer is $-1$

for $3)$ I do not get the answer, can you solve for me, so I relate, where I did mistake?

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3) 4/17 ??

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Yes, can you explain a bit?

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this is a rough work...hope you understand easily

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thanks

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megma ..why u took eigen value 1 for vector [1 2] and -2 for vector [9 1] ?

is there any rule ?

i guess there will be 4 combination of ax=eigen value*x . for which correct combination will be as u mention becoz these follow e1+e2= -1 and e1*e2= detA

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read it again

Eigen value are 1,−2 and Corresponding Eigen Vectors

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my bad (y)

thanks

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I have one more method for $3)A=MDM^{-1}$(This is called concept of diagonalization) where $M$ is the model matrix(Collection of eigen vectors) like this $M=\begin{bmatrix} 1 &9 \\ 2 &1 \end{bmatrix}$ and $D$ is is called Diagonal matrix(collection of eigen values) like this $D=\begin{bmatrix} 1 &0 \\ 0 &-2 \end{bmatrix}$ and $M^{-1}=\frac{adj(M)}{|M|}$

Answer 1

1. Sum of elements when i = j, (- 37 / 17) + (20 / 17) = (- 17 / 17) = -1

2. Det (A) = (- 37 / 17) * ( 20 / 17) - ( 27 / 17) * ( -6 / 17) = (- 578 / 17) = -34

3. Sum of all the elements of the matrix  = (- 37 / 17) + (27 / 17) + (- 6 / 17) + (20 / 17) = (- 43 / 17) + (47 / 17) = 4 / 17