Suppose A is any 3$$ \times $$ 3 non-singular matrix and ( A $$-$$ 3I) (A $$-$$ 5I) = O where I = I3 and O = O3. If $$\alpha $$A + $$\beta $$A-1 = 4I, then $$\alpha $$ + $$\beta $$ is equal to :
JEE · Math · previous-year question
- A.8correct
- B.7
- C.13
- D.12
Answer
A. 8
Explanation
Given, ( A $$-$$ 3I) (A $$-$$ 5I) = O $$ \Rightarrow $$ A2 - 8A + 15I = O Multiplying both sides by A- 1, we get, A- 1A.A - 8A- 1A + 15A- 1I = A- 1O $$ \Rightarrow $$ A - 8I + 15A- 1 = O $$ \Rightarrow $$ A + 15A- 1 = 8I $$ \Rightarrow $$$${A \over 2} + {{15{A^{ - 1}}} \over 2} = 4I$$ Comparing with the equation $$\alpha $$A + $$\beta $$A-1 = 4I, we get $$\alpha $$ = $${1 \over 2}$$ and $$\beta $$ = $${15 \over 2}$$ $$ \therefore $$ $$\alpha $$ + $$\beta $$ = $${1 \over 2}$$ + $${15 \over 2}$$ = $${16 \over 2}$$ = 8
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