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NDA 2020
Exam
NDA
Year
2020
Subject
Mathematics
Chapter
Matrices

NDA 2020 · Mathematics · Matrices

Let A= [ array cc x+y & y 2 x & x-y array ], B= [ array c 2 -1...

Let A=A=\left[x+yy2xx−y\begin{array}{cc}x+y & y \\ 2 x & x-y\end{array}\right], B=\left[2−1\begin{array}{c}2 \\ -1\end{array}\right] and C=\left[32\begin{array}{l}3 \\ 2\end{array}\right]. If AB=CA B=C, then what is the value of the determinant of the matrix AA ?

Options

  1. A-10
  2. B-14
  3. C-24
  4. D-34

Correct answer

B. -14

Step-by-step solution

AB=[x+yy2xx−y][2−1]=[2(x+y)−y4x−x+y]=[2x+y3x+y]\begin{aligned} A B & =\left[\begin{array}{cc} x+y & y \\ 2 x & x-y \end{array}\right]\left[\begin{array}{c} 2 \\ -1 \end{array}\right] \\ & =\left[\begin{array}{c} 2(x+y)-y \\ 4 x-x+y \end{array}\right]=\left[\begin{array}{l} 2 x+y \\ 3 x+y \end{array}\right] \end{aligned} As AB=CA B=C∴[2x+y3x+y]=[32]2x+y=3(i)\begin{aligned} & \therefore\left[\begin{array}{l} 2 x+y \\ 3 x+y \end{array}\right]=\left[\begin{array}{l} 3 \\ 2 \end{array}\right] \\ & 2 x+y=3 (i) \end{aligned}andand3 x+y=2(ii)Fromequation(i)and(ii),weget(ii) From equation (i) and (ii), we getx=-1, y=5..∴A=[−1+55−2−1−5]=[45−2−6]=4(−6)−5(−2)=−14\begin{aligned} \therefore A & =\left[\begin{array}{cc}-1+5 & 5 \\ -2 & -1-5\end{array}\right]=\left[\begin{array}{cc}4 & 5 \\ -2 & -6\end{array}\right] \\ & =4(-6)-5(-2)=-14\end{aligned}$
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