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 $\left(-1{\right)}^{i+j}$, 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 ${C}_{ij}=\left(-1{\right)}^{i+j}×{M}_{ij}$, where ${M}_{ij}$ is the minor of ${a}_{ij}$.

Properties and Significance:

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

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

3. Matrix Properties: Minors and cofactors provide insights into the properties of matrices, especially regarding invertibility and determinant properties.

Example:

Consider a $3×3$ matrix $A$:

$A=\left[\begin{array}{ccc}2& 3& 1\\ 5& 4& 6\\ 0& 7& 2\end{array}\right]$
1. 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.

2. The cofactor ${C}_{21}$of the element ${a}_{21}$is obtained by multiplying the minor ${M}_{21}$ by $\left(-1{\right)}^{2+1}$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.