Let A be a 2 × 2 symmetric matrix such that A [ array l 1 1 array ]= [ array l 3 7 array ] and the determinant of A be 1 . If A^ -1 =α A+β I , where I is an identity matrix of order 2 × 2 , then α+β equals _______

Correct answer: 0 aligned & Let A=[ array ll a & b b & d array ] & [ array ll a & b b & d array ][ array l 1 1 array ]=[ array l 3 7 array ], a d-b^2=1 & a+b=3, b+d=7,(3-b)(7-b)-b^2=1 & 21-10 b=1 arrow b=2, a=1, d=5 & A=[ array ll 1 & 2 2 & 5 array ], A^ -1 =[ array cc 5 & -2 -2 & 1 array ] & A^ -1 =α A+β I & [ array cc 5 & -2 -2 & 1 array ]=[ array cc α+β & 2 α 2 α & 5 α+β array ] & α=-1, β=6 arrow α+β=5 aligned

JEE Main 20244 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 4 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2024 · 4 April · Evening Shift · Mathematics · Matrices

Let A be a 2 × 2 symmetric matrix such that A [ array l 1 1 array...

Let

A

be a

2 \times 2

symmetric matrix such that

A

\left[

\begin{array}{l}1 \\ 1\end{array}

\right]=

\left[

\begin{array}{l}3 \\ 7\end{array}

\right]

and the determinant of

A

be 1 . If

A^{-1}=\alpha A+\beta I

, where

I

is an identity matrix of order

2 \times 2

, then

\alpha+\beta

equals _______

Correct answer

Step-by-step solution

\begin{aligned} & \text{ Let } A=\left[\begin{array}{ll}a & b \\ b & d\end{array}\right] \\ & {\left[\begin{array}{ll}a & b \\ b & d\end{array}\right]\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right], a d-b^2=1} \\ & a+b=3, b+d=7,(3-b)(7-b)-b^2=1 \\ & 21-10 b=1 \rightarrow b=2, a=1, d=5 \\ & A=\left[\begin{array}{ll}1 & 2 \\ 2 & 5\end{array}\right], A^{-1}=\left[\begin{array}{cc}5 & -2 \\ -2 & 1\end{array}\right] \\ & A^{-1}=\alpha A+\beta I \\ & {\left[\begin{array}{cc}5 & -2 \\ -2 & 1\end{array}\right]=\left[\begin{array}{cc}\alpha+\beta & 2 \alpha \\ 2 \alpha & 5 \alpha+\beta\end{array}\right]} \\ & \alpha=-1, \beta=6 \rightarrow \alpha+\beta=5\end{aligned}

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