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

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(D) Given that $(A-3 I)(A-5 I)=O$. Expanding the expression,we get: $A^2 - 5A - 3A + 15I = O$ $A^2 - 8A + 15I = O$ $A^2 + 15I = 8A$ Multiplying both s

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$8$

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(D) Given that $(A-3 I)(A-5 I)=O$. Expanding the expression,we get: $A^2 - 5A - 3A + 15I = O$ $A^2 - 8A + 15I = O$ $A^2 + 15I = 8A$ Multiplying both sides by $A^{-1}$,we get: $A + 15A^{-1} = 8I$ Dividing the entire equation by $2$: $\frac{1}{2}A + \frac{15}{2}A^{-1} = 4I$ Comparing this with the given equation $\alpha A + \beta A^{-1} = 4I$,we identify: $\alpha = \frac{1}{2}$ 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 $(A-3 I)(A-5 I)=O$,where $I=I_3$ and $O=O_3$. Here $O_3$ represents the zero matrix of order $3$ and $I_3$ is the identity matrix of order $3$. If $\alpha A+\beta A^{-1}=4 I$,then $\alpha+\beta$ is equal to:

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