Question

The adjoint of the matrix

\[ \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \]

is:

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

Hint:

The adjoint of a matrix is found by taking the transpose of its cofactor matrix.

Answer 1

The Correct Option is C

Step 1: Finding the adjoint of a matrix.

The adjoint of a matrix is the transpose of its cofactor matrix.
Given \[ A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \] we first compute its cofactors.

Step 2: Computing the cofactors.

\[ C_{11} = (+1)\det[4] = 4 \] \[ C_{12} = (-1)\det[2] = -2 \] \[ C_{21} = (-1)\det[3] = -3 \] \[ C_{22} = (+1)\det[1] = 1 \]

Thus, the cofactor matrix is:

\[ C = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \]

Step 3: Adjoint of the matrix.

The adjoint of \( A \) is the transpose of the cofactor matrix:

\[ \operatorname{adj}(A) = C^T = \begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix} \]

Step 4: Conclusion.

The adjoint of the matrix is:

\[ \begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix} \]