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The Eigen Vectors of the Matrix  $A=\begin{bmatrix} 3 &4 \\ 4 &-3 \end{bmatrix}$ are $\begin{bmatrix} a\\ 1 \end{bmatrix},\begin{bmatrix} 1\\ b \end{bmatrix}$ the $a+b=?$

Answer is given (a+b)=0 how ?????
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For Eigen Values:

Let us suppose eigenvalues are $\lambda_{1},\lambda_{2}$

Sum of all Eigen Values$=$Trace of the matrix=Sum of Leading diagonal element

$\lambda_{1}+\lambda_{2}=0$------->(1)

Product of all diagonal element$=Det(A)=|A|$

$\lambda_{1}.\lambda_{2}=-9-16$

$\lambda_{1}.\lambda_{2}=-25$------>(2)

Now ,we can make Quadratic Equation if given roots are $\lambda_{1},\lambda_{2}$

$x^{2}-(\lambda_{1}+\lambda_{2})x+\lambda_{1}.\lambda_{2}=0$

put the value from the abpve equation $(1)$ and $(2)$,and we get

$x^{2}-0.x-25=0$

$x^{2}-25=0$

$x^{2}=25$

$x=5,-5$

So$,\lambda_{1}=-5,\lambda_{2}=5$

Find the eigen values using det(A-$\Lambda$I)=0 it'll be 5,-5

Then solve for value of a and b using (A-$\Lambda$I)x=0 where x are two given eigen vectors, you will get value of a,b as a=2 b=-2

It's very easy maybe you are doing some calculation mistake.
answered by Active (2.3k points)
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i am  getting same answer but still doubts will solving thanks

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