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NDA 2019 Maths · Mathematics · Matrices · medium

If A=beginpmatrix1&-1\\1&1endpmatrix, then A^4 equals

Single correct option

If A=(1−111)A=\begin{pmatrix}1&-1\\1&1\end{pmatrix}, then A4A^4 equals
  1. A. 4I4I
  2. B. 16I16I
  3. C. −2I-2I
  4. D. −4I-4I
Show answer and solution

Answer: (D) −4I-4I

Compute A2=(0−220)A^2=\begin{pmatrix}0&-2\\2&0\end{pmatrix}. Then A4=(A2)2=(−400−4)=−4IA^4=(A^2)^2=\begin{pmatrix}-4&0\\0&-4\end{pmatrix}=-4I. The sign is negative because the product of the two off-diagonal entries is −2⋅2=−4-2\cdot2=-4. Raising each entry to the fourth power, or using (2)4=4(\sqrt2)^4=4 without the rotation sign, leads to 4I4I.
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Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake