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  4. If A is nonsingular and (A - 2I) (A - 4I) = O, then (1/6)A + (4/3)A-1 is equal to kI. Then the value of k is .

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Q. If $A$ is nonsingular and $(A - 2I) (A - 4I) = O$, then $\frac{1}{6}A + \frac{4}{3}A^{-1}$ is equal to $kI$. Then the value of $k$ is ______.

Matrices

Solution:

We have,
$(A - 2I)(A - 4I) = 0$
or $A^2 - 2A - 4A + 8I = O$
or $A^2 - 6A + 8I = O$
or $A^{-1}(A^2 - 6A + 8I) = A^{-1}O$
or $A - 6I + 8A^{-1} = O$
or $A + 8A^{-1} = 6I$
or $\frac{1}{6}A + \frac{4}{3} A^{-1} = 1$