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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Matrices

NDA 2019 · Mathematics · Matrices

If A=[ array cc 1 & -1 -1 & 1 array ] , then the expression A^3-2...

If A=\left[1−1−11\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right], then the expression A3−2A2\mathrm{A}^3-2 \mathrm{ A}^2 is

Options

  1. Aa null matrix
  2. Ban identity matrix
  3. Cequal to A
  4. Dequal to -A

Correct answer

A. a null matrix

Step-by-step solution

A=[1−1−11]∴A2=[1−1−11][1−1−11]=[2−2−22]=2A∴A3−2A2=A⋅A2−2A2=A(2A)−2(2A)=2A2−4A=2(2A)−4A=4A−4A=0\begin{aligned} & A=\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right] \\ & \therefore A^2=\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right] \\ & =\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]=2 A \\ & \therefore A^3-2 A^2=A \cdot A^2-2 A^2 \\ & =A(2 A)-2(2 A) \\ & =2 A^2-4 A \\ & =2(2 A)-4 A \\ & =4 A-4 A \\ & =0\end{aligned}
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