Question

Find the adjoint of \[ A= \begin{pmatrix} 1& 2\\ 3& 4 \end{pmatrix} \]

• A. \[ \begin{pmatrix} 4&-2\\ -3& 1 \end{pmatrix} \]
• B. \[ \begin{pmatrix} 1&3\\ 2& 4 \end{pmatrix} \]
• C. \[ \begin{pmatrix} 4& 2\\ 3& 1 \end{pmatrix} \]
• D. \[ \begin{pmatrix} -4& 2\\ 3&-1 \end{pmatrix} \]

Hint:

For \(2\times2\) matrices: Swap diagonal entries and change signs of off-diagonal entries.

Answer 1

The Correct Option is A

Concept: For a \(2\times2\) matrix: \[ A= \begin{pmatrix} a& b\\ c& d \end{pmatrix} \] the adjoint is: \[ \operatorname{adj}(A)= \begin{pmatrix} d&-b\\ -c& a \end{pmatrix} \]

Step 1: Identify matrix entries Given: \[ A= \begin{pmatrix} 1& 2\\ 3& 4 \end{pmatrix} \] Hence: \[ a=1,\quad b=2,\quad c=3,\quad d=4 \]

Step 2: Apply adjoint formula Swap diagonal elements: \[ 1\leftrightarrow4 \] Change signs of off-diagonal elements: \[ 2\to-2 \] \[ 3\to-3 \] Thus: \[ \operatorname{adj}(A)= \begin{pmatrix} 4&-2\\ -3& 1 \end{pmatrix} \] Final Answer: \[ \boxed{ \begin{pmatrix} 4&-2\\ -3& 1 \end{pmatrix} } \]