Matrices and Determinants jee mains questions | JEE Main Maths Matrices and Determinants Previous Year Question

Q 21 :

If $A = [\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}],$ then

$A^{'} A = I$
$A^{'} A = 0$
$A^{'} A = 2 I$
$A^{'} A = - I$

(1)

Ans. $A^{'} A = I$

Explanation:

Given, $A = [\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}]$

$A^{'} = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}]$

$A^{'} A = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}] [\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}]$ $= [\begin{matrix} \cos^{2} α + \sin^{2} α & \sin α \cos α - \cos α \sin α \\ \cos α \sin α - \sin α \cos α & \sin^{2} α + \cos^{2} α \end{matrix}]$

$= [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = I$

$∴$ $A^{'} A = I$

Q 22 :

If $A = [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}],$ then $A^{2} - 5 A + 7 I =$

$0$
$I$
$2 I$
$3 I$

(1)

Ans. 0

Explanation:

$A = [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}]$

$A^{2} - 5 A + 7 I = ?$

$A^{2} = [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}] [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}] = [\begin{matrix} 8 & 5 \\ - 5 & 3 \end{matrix}]$

$- 5 A = - 5 [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}] = [\begin{matrix} - 15 & - 5 \\ 5 & - 10 \end{matrix}]$

$7 I = 7 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}]$

$A^{2} - 5 A + 7 I = [\begin{matrix} 8 & 5 \\ - 5 & 3 \end{matrix}] + [\begin{matrix} - 15 & - 5 \\ 5 & - 10 \end{matrix}] + [\begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}]$

$= [\begin{matrix} 8 - 15 + 7 & 5 - 5 + 0 \\ - 5 + 5 + 0 & 3 - 10 + 7 \end{matrix}] = 0$

Topic Question Set

Q 21 :

If $A = [\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}],$ then

Q 22 :

If $A = [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}],$ then $A^{2} - 5 A + 7 I =$

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Topic Question Set

Q 21 : If A=[cosαsinα-sinαcosα], then

Q 22 : If A=[31-12], then A2-5A+7I=

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Q 21 :

If $A = [\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}],$ then

Q 22 :

If $A = [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}],$ then $A^{2} - 5 A + 7 I =$