Question

Find the minors and cofactor of the element determinant $\begin{vmatrix} 2 & -3 \\ 4 & 7 \end{vmatrix}$

Answer 1

M11 = 7, M12 = 4, M21 = -3, M22 = 2; C11 = 7, C12 = -4, C21 = 3, C22 = 2

Answer 2

For a $2 \times 2$ matrix, the minor $M_{ij}$ of an element is the determinant of the $1 \times 1$ matrix that remains after removing the $i^\text{th}$ row and $j^\text{th}$ column.

• Element $a_{11}=2$:
  $M_{11} = 7$.
  Cofactor: $C_{11} = (-1)^{1+1} M_{11} = 7$.

• Element $a_{12}=-3$:
  $M_{12} = 4$.
  Cofactor: $C_{12} = (-1)^{1+2} M_{12} = -4$.

• Element $a_{21}=4$:
  $M_{21} = -3$.
  Cofactor: $C_{21} = (-1)^{2+1} M_{21} = 3$.

• Element $a_{22}=7$:
  $M_{22} = 2$.
  Cofactor: $C_{22} = (-1)^{2+2} M_{22} = 2$.

Minors:

$$M_{11} = 7,\quad M_{12} = 4,\quad M_{21} = -3,\quad M_{22} = 2.$$

Cofactors:

$$C_{11} = 7,\quad C_{12} = -4,\quad C_{21} = 3,\quad C_{22} = 2.$$