QJEEM20Mafa31: JEE Main 2020 Mathematics - Class 12 JEEM20Mafa31 Solution: If A = ( 2 2 9 4 ) and I = ( 1 0 0 1 ) then 10A –1 is equal to :

Question

Verified Solution • 2025–26 Session moderate

If

A = (\begin{matrix} 2 & 2 \\ 9 & 4 \end{matrix})

and

I = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

then 10A^–1 is equal to :

Accepted Answer

According to Cayley Hamilton equation
|A –

\lambda

I| = 0

\Rightarrow

| \begin{matrix} 2 - λ & 2 \\ 9 & 4 - λ \end{matrix} |

= 0

\Rightarrow

(2 –

\lambda

)(4 –

\lambda

) – 18 = 0

\Rightarrow

8 – 2

\lambda

– 4

\lambda

+

\lambda

² – 18 = 0

\Rightarrow

\lambda

² – 6

\lambda

– 10 = 0

\therefore

A² – 6A– 10 = 0

\Rightarrow

A^–1(A²) – 6A^–1A – 10A^–1 = 0

\Rightarrow

A – 6I – 10A^–1 = 0

\Rightarrow

10A^–1 = A – 6I

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