Minors and Cofactors of Determinants
Minors:
 Definition: The minor of an element in a matrix is the determinant of the smaller matrix obtained by removing the row and column containing that element.
For a matrix $A$ with elements ${a}_{ij}$, the minor of ${a}_{ij}$, denoted as ${M}_{ij}$, is the determinant of the matrix formed by deleting the $i$th row and $j$th column.
Cofactors:
 Definition: Cofactors are scalar values associated with each element of a matrix, obtained by multiplying the minor of the element by $(1{)}^{i+j}$, where $i$ and $j$ are the row and column indices, respectively.
For a matrix $A$ with elements ${a}_{ij}$, the cofactor of ${a}_{ij}$, denoted as ${C}_{ij}$, is given by ${C}_{ij}=(1{)}^{i+j}\times {M}_{ij}$, where ${M}_{ij}$ is the minor of ${a}_{ij}$.
Properties and Significance:

Determinant Calculation: Cofactors play a crucial role in calculating the determinant of a matrix using the formula $\text{det}(A)={\sum}_{j=1}^{n}{a}_{ij}\times {C}_{ij}$ (expanding along any row or column).

Adjoint and Inverse Matrices: Cofactors are used in the calculation of the adjoint matrix and subsequently in finding the inverse of a matrix.

Matrix Properties: Minors and cofactors provide insights into the properties of matrices, especially regarding invertibility and determinant properties.
Example:
Consider a $3\times 3$ matrix $A$:

The minor ${M}_{21}$(the minor of the element in the second row, first column) is obtained by removing the second row and first column and calculating its determinant.

The cofactor ${C}_{21}$of the element ${a}_{21}$is obtained by multiplying the minor ${M}_{21}$ by $(1{)}^{2+1}$because the row index + column index is odd).
Applications:

Determinants: Minors and cofactors are essential in expanding determinants by minors or cofactors.

Matrix Properties: They contribute to understanding matrix properties, including invertibility, adjoint calculation, and system solvability.