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Suppose A is any 3×3 non-singular matrix and (A−3I)(A−5I)=O, where I=I3 and O=O3. If αA+βA−1=4I, Then \alpha +\beta is equal to :

  • Option 1)

    8

  • Option 2)

    7

  • Option 3)

    13

  • Option 4)

    12

 

Answers (2)

best_answer

As we learned

 

Inverse of a matrix -

A non-singular square matrix of order n is invertible if there exists a square matrix B of the same order such that AB=I=BA

-

 

 

\left ( A-3I \right )\left ( A-5I \right )=0

\Rightarrow \: A^{2}-8A+15I=0\cdots \cdots \left ( i \right )

Also given that  \alpha A+\beta A^{-1}=4I

Multiply both sides by A

\Rightarrow\: \alpha A^{2}-4A+\beta I=0\cdots \cdots \left ( ii \right )

Compare (i) and (ii)

A^{2}-8A+15I=0

\alpha A^{2}-4A+\beta I=0

Thus \frac{1}{\alpha }=\frac{8}{4}=\frac{15}{\beta }

Thus \alpha =\frac{1}{2}   and \beta =\frac{15}{2}

\Rightarrow \: \alpha +\beta =8

 


Option 1)

8

Option 2)

7

Option 3)

13

Option 4)

12

Posted by

Himanshu

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