JEE Main 2024 — Mathematics Algebra

Let, $A=[\begin{matrix}{x}_{1} & {y}_{1} & {z}_{1} \\ {x}_{2} & {y}_{2} & {z}_{2} \\ {x}_{3} & {y}_{3} & {z}_{3}\end{matrix}]$A=[x1​x2​x3​​y1​y2​y3​​z1​z2​z3​​]

It is given that, $A[\begin{matrix}1 \\ 0 \\ 1\end{matrix}]=[\begin{matrix}2 \\ 0 \\ 2\end{matrix}]$A[101​]=[202​]

$$\Rightarrow [\begin{matrix}{x}_{1}+{z}_{1} \\ {x}_{2}+{z}_{2} \\ {x}_{3}+{z}_{3}\end{matrix}]=[\begin{matrix}2 \\ 0 \\ 2\end{matrix}]$$⇒[x1​+z1​x2​+z2​x3​+z3​​]=[202​]

$$\Rightarrow {x}_{1}+{z}_{1}=2,{x}_{2}+{z}_{2}=0,{x}_{3}+{z}_{3}=0...(i)$$⇒x1​+z1​=2,x2​+z2​=0,x3​+z3​=0...(i)

Also, $A[\begin{matrix}-1 \\ 0 \\ 1\end{matrix}]=[\begin{matrix}-4 \\ 0 \\ 4\end{matrix}]$A[−101​]=[−404​]

$$\Rightarrow [\begin{matrix}-{x}_{1}+{z}_{1} \\ -{x}_{2}+{z}_{2} \\ -{x}_{3}+{z}_{3}\end{matrix}]=[\begin{matrix}-4 \\ 0 \\ 4\end{matrix}]$$⇒[−x1​+z1​−x2​+z2​−x3​+z3​​]=[−404​]

$$\Rightarrow -{x}_{1}+{z}_{1}=-4,-{x}_{2}+{z}_{2}=0,-{x}_{3}+{z}_{3}=4...(ii)$$⇒−x1​+z1​=−4,−x2​+z2​=0,−x3​+z3​=4...(ii)

Now, $A[\begin{matrix}0 \\ 1 \\ 0\end{matrix}]=[\begin{matrix}0 \\ 2 \\ 0\end{matrix}]$A[010​]=[020​]

$$\Rightarrow [\begin{matrix}{y}_{1} \\ {y}_{2} \\ {y}_{3}\end{matrix}]=[\begin{matrix}0 \\ 2 \\ 0\end{matrix}]$$⇒[y1​y2​y3​​]=[020​]

$$\Rightarrow {y}_{1}=0,{y}_{2}=2,{y}_{3}=0...(iii)$$⇒y1​=0,y2​=2,y3​=0...(iii)

Using $(i)\text{and}(ii)$(i)and(ii),

$$\Rightarrow {x}_{1}=3,{x}_{2}=0,{x}_{3}=-1$$⇒x1​=3,x2​=0,x3​=−1

$$\Rightarrow {z}_{1}=-1,{z}_{2}=0,{z}_{3}=3$$⇒z1​=−1,z2​=0,z3​=3

$$\therefore A=[\begin{matrix}3 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0 & 3\end{matrix}]$$∴A=[30−1​020​−103​]

Now, $(A-3I)[\begin{matrix}x \\ y \\ z\end{matrix}]=[\begin{matrix}-1 \\ 2 \\ 3\end{matrix}]$(A−3I)[xyz​]=[−123​]

$$\Rightarrow [\begin{matrix}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{matrix}][\begin{matrix}x \\ y \\ z\end{matrix}]=[\begin{matrix}-1 \\ 2 \\ 3\end{matrix}]$$⇒[00−1​0−10​−100​][xyz​]=[−123​]

$$\Rightarrow [\begin{matrix}-z \\ -y \\ -x\end{matrix}]=[\begin{matrix}1 \\ 2 \\ 3\end{matrix}]$$⇒[−z−y−x​]=[123​]

$$\Rightarrow z=-1,y=-2,x=-3$$⇒z=−1,y=−2,x=−3