Suppose A is any 3 times 3 non-singular matri | vedclass

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(A) Given the characteristic equation $(A - 3I)(A - 5I) = O$. Expanding this,we get $A^2 - 8A + 15I = O$. Since $A$ is non-singular,we can multiply th

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$8$

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(A) Given the characteristic equation $(A - 3I)(A - 5I) = O$. Expanding this,we get $A^2 - 8A + 15I = O$. Since $A$ is non-singular,we can multiply throughout by $A^{-1}$: $A^{-1}(A^2 - 8A + 15I) = A^{-1}O$ $A - 8I + 15A^{-1} = O$ $A + 15A^{-1} = 8I$. To match the form $\alpha A + \beta A^{-1} = 4I$,we divide the entire equation by $2$: $\frac{1}{2}A + \frac{15}{2}A^{-1} = 4I$. Comparing this with $\alpha A + \beta A^{-1} = 4I$,we find $\alpha = \frac{1}{2}$ and $\beta = \frac{15}{2}$. Therefore,$\alpha + \beta = \frac{1}{2} + \frac{15}{2} = \frac{16}{2} = 8$.

Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A - 3I)(A - 5I) = O$,where $I = I_3$ and $O = O_3$. If $\alpha A + \beta A^{-1} = 4I$,then $\alpha + \beta$ is equal to

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