Which of the following is the formula for cofactor of an element aij ?

The correct option is Aᵢⱼ=(-1)ᶦ⁺ʲ Mᵢⱼ.

To understand why this is the correct formula, let us break down the definition of minors and cofactors in linear algebra.

For a given square matrix, the minor of an element $a_{ij}$, denoted as $M_{ij}$, is the determinant of the submatrix obtained by deleting the $i$-th row and $j$-th column.

The cofactor of the element $a_{ij}$, denoted by $A_{ij}$ (or sometimes $C_{ij}$), is the signed minor. The sign is determined by the position of the element in the matrix, specifically based on its row index $i$ and column index $j$.

The sign factor is calculated as:
${(-1)}^{i+j}$

If the sum of the row and column indices ($i+j$) is even, the sign is positive ($+1$). If the sum ($i+j$) is odd, the sign is negative ($-1$).

Combining the sign factor with the minor gives us the standard mathematical formula for the cofactor of an element:
$A_{ij}={(-1)}^{i+j}{M}_{ij}$

This corresponds directly to the option Aᵢⱼ=(-1)ᶦ⁺ʲ Mᵢⱼ, where the superscript exponents are written as subscripts/superscripts in the text representation.