Question

Find the cofactors of the elements of the matrix \( \begin{bmatrix} -1 & 2 \\ -3 & 4 \end{bmatrix} \)

Hint:

Cofactor \( C_{ij} = (-1)^{i+j} \times \text{minor} \); for 2×2 matrix, minor is the opposite diagonal element.

Answer 1

For a 2×2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), cofactors are: 

\( C_{11} = (-1)^{1+1} \cdot d = d \), \( C_{12} = (-1)^{1+2} \cdot c = -c \), 

\( C_{21} = (-1)^{2+1} \cdot b = -b \), \( C_{22} = (-1)^{2+2} \cdot a = a \). 

For \( \begin{bmatrix} -1 & 2 \\ -3 & 4 \end{bmatrix} \): 

- \( C_{11} = (-1)^{1+1} \cdot 4 = 4 \). 

- \( C_{12} = (-1)^{1+2} \cdot (-3) = 3 \). 

- \( C_{21} = (-1)^{2+1} \cdot 2 = -2 \). 

- \( C_{22} = (-1)^{2+2} \cdot (-1) = -1 \). 

Answer: Cofactors: \( C_{11} = 4 \), \( C_{12} = 3 \), \( C_{21} = -2 \), \( C_{22} = -1 \).