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NDA 2023 Maths · Mathematics · Matrices · medium

If A=beginpmatrix2&3\\1&2endpmatrix, then A^-1 equals

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If A=(2312)A=\begin{pmatrix}2&3\\1&2\end{pmatrix}, then A−1A^{-1} equals
  1. A. (2−3−12)\begin{pmatrix}2&-3\\-1&2\end{pmatrix}
  2. B. (2312)\begin{pmatrix}2&3\\1&2\end{pmatrix}
  3. C. (−231−2)\begin{pmatrix}-2&3\\1&-2\end{pmatrix}
  4. D. 12(2−3−12)\dfrac12\begin{pmatrix}2&-3\\-1&2\end{pmatrix}
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Answer: (A) (2−3−12)\begin{pmatrix}2&-3\\-1&2\end{pmatrix}

∣A∣=4−3=1|A|=4-3=1, and for a 2×22\times2 matrix A−1=1∣A∣(d−b−ca)A^{-1}=\frac1{|A|}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}, so swap the diagonal entries and negate the off-diagonal ones: (2−3−12)\begin{pmatrix}2&-3\\-1&2\end{pmatrix}. Negating the diagonal instead, or dividing by 22, are the traps.
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