1.

The inverse of the 2 x 2 matrix \(\begin{bmatrix} 2 & 3\\ 4 & 1 \end{bmatrix}\) is1. \( \frac{1}{10}\begin{bmatrix} 1 & -3\\ -4 & 2 \end{bmatrix}\)2. \( \frac{1}{10}\begin{bmatrix} 1 & 3\\ 4 & 2 \end{bmatrix}\)3. \( \frac{1}{10}\begin{bmatrix} -1 & -3\\ -4 & -2 \end{bmatrix}\)4. \( \frac{1}{10}\begin{bmatrix} -1 & 3\\ 4 & -2 \end{bmatrix}\)

Answer» Correct Answer - Option 4 : \( \frac{1}{10}\begin{bmatrix} -1 & 3\\ 4 & -2 \end{bmatrix}\)

Concept:

For a 2 x 2 matrix there is a short-cut formula for inverse as given 

\(\rm {\begin{bmatrix} a & & b\\ c& & d \end{bmatrix}}^{-1} = \frac{1}{(ad-bc)}\begin{bmatrix} d & & -b\\ -c& & a \end{bmatrix}\)

Calculation:

As we know that inverse of matrix \(\rm {\begin{bmatrix} a & & b\\ c& & d \end{bmatrix}}^{-1} = \frac{1}{(ad-bc)}\begin{bmatrix} d & & -b\\ -c& & a \end{bmatrix}\)

⇒\({\begin{bmatrix} 2 & 3\\ 4 & 1 \end{bmatrix}}^{-1}= \frac{1}{(2.1-4.3)}\begin{bmatrix} 1 & -3\\ -4 & 2 \end{bmatrix}\)

⇒ \({\begin{bmatrix} 2 & 3\\ 4 & 1 \end{bmatrix}}^{-1}= \frac{1}{-10}\begin{bmatrix} 1 & -3\\ -4 & 2 \end{bmatrix}\)

⇒ \({\begin{bmatrix} 2 & 3\\ 4 & 1 \end{bmatrix}}^{-1}= \frac{1}{10}\begin{bmatrix} -1 & 3\\ 4 & -2 \end{bmatrix}\)

Hence option 4 is the correct answer.


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