QJEEM18Ma0717: JEE Main 2018 Mathematics - Class 12 JEEM18Ma0717 Solution: Suppose A is any 3 × 3 non-singular matrix and ( A − 3I) (A −…

Question

Verified Solution • 2025–26 Session moderate

Suppose A is any 3

\times

3 non-singular matrix and ( A

-

3I) (A

-

5I) = O where I = I₃ and O = O₃. If

α

A +

β

A^-1 = 4I, then

α

+

β

is equal to :

Accepted Answer

Given,

( A

-

3I) (A

-

5I) = O

\Rightarrow

A² - 8A + 15I = O

Multiplying both sides by A^{- 1}, we get,

A^{- 1}A.A - 8A^{- 1}A + 15A^{- 1}I = A^{- 1}O

\Rightarrow

A - 8I + 15A^{- 1} = O

\Rightarrow

A + 15A^{- 1} = 8I

\Rightarrow

\frac{A}{2} + \frac{15 A^{- 1}}{2} = 4 I

Comparing with the equation

α

A +

β

A^-1 = 4I, we get

α

=

\frac{1}{2}

and

β

=

\frac{15}{2}

∴

α

+

β

=

\frac{1}{2}

+

\frac{15}{2}

=

\frac{16}{2}

= 8

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