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The adjoint of matrix \(\left[ {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right]\)is

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TNUSRB SI 2020 Official Paper (Held on 12 January 2020)
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  1. \(\left[ {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right]\)
  2. \(\left[ {\begin{array}{*{20}{c}} -a&-b\\ -c&-d \end{array}} \right]\)
  3. \(\left[ {\begin{array}{*{20}{c}} d&-b\\ -c&a \end{array}} \right]\)
  4. \(\left[ {\begin{array}{*{20}{c}} d&b\\ c&a \end{array}} \right]\)

Answer (Detailed Solution Below)

Option 3 : \(\left[ {\begin{array}{*{20}{c}} d&-b\\ -c&a \end{array}} \right]\)
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Detailed Solution

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Explanation:
 
1. Start with a 2x2 matrix A:
 
       A = [a  c]
             [b  d]
 
2. The adjoint (also called the adjugate) of a matrix is found by swapping the elements on the main diagonal and changing the signs of the elements on the other diagonal. This can be expressed as a formula:
 
       adj(A) = [d  -c]
                     [-b  a]
The correct answer is option 3.
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