Example $8$:$59$ If $A$ is non$-$singular, prove that the eigenExample $9\mathrm{.}55$ The chances of defective screws in three

boxes $A,B,C$ are $\frac{1}{5},\frac{1}{6},\frac{1}{7},$ respectively. A box is selected

at random and a screw drawn from it at random is found

to be defective. Then, find the probability that it came from

box $A.$

values of $A^{-1}$ are the reciprocals of the eigen value of $A.$

Sol. Let $A$ be an eigen value of A and $x$ be a corresponding

eigenvector. Then,