Minor of a matrix
Consider a $3 \times 3$ matrix $A$ and a row number $i$ and a column number $j$. The $(i, j)$ minor of $A$ is the determinant of the $2 \times 2$ matrix that is left after crossing out row $i$ and column $j$ of $A$.
For example, consider the matrix
\[A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & k \end{bmatrix}.\]Crossing out row $2$ and column $3$ of $A$ leaves the $2 \times 2$ matrix with the entries $a$, $b$, $g$, and $h$, so the $(2, 3)$ minor of $A$ is
\[M_{23} = \det \begin{bmatrix} a & b \\ g & h \end{bmatrix} = a h - b g.\]Consider an $n \times n$ matrix $A$ and indices $i$ and $j$, and let $A^{(i, j)}$ be the $(n - 1) \times (n - 1)$ matrix obtained by deleting row $i$ and column $j$ of $A$. The $(i, j)$ minor of $A$ is the determinant
\[M_{ij} = \det A^{(i, j)}.\]Examples
Worked examples
-
The matrix \(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{bmatrix}\)
: $M_{11} = 2$, $M_{12} = -2$, $M_{13} = -3$
Undergraduate
The minors for the first row of the matrix \[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{bmatrix}\] are the determinants of the $2 \times 2$ matrices left after deleting row $1$ and one column: \[M_{11} = \det \begin{bmatrix} 5 & 6 \\ 8 & 10 \end{bmatrix} = 2, \quad M_{12} = \det \begin{bmatrix} 4 & 6 \\ 7 & 10 \end{bmatrix} = -2, \quad M_{13} = \det \begin{bmatrix} 4 & 5 \\ 7 & 8 \end{bmatrix} = -3.\]
-
The matrix \(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\)
: $M_{11} = 3$, $M_{12} = 4$, $M_{21} = 1$, $M_{22} = 2$
Undergraduate
Each minor of \(A = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) is the single entry left after deleting one row and one column; for example, deleting row $1$ and column $2$ leaves $4$, so $M_{12} = 4$.
Properties
- The $(i, j)$ cofactor of $A$ is the $(i, j)$ minor with a sign: $C_{ij} = (-1)^{i + j} M_{ij}$.
- A symmetric matrix is positive definite if and only if all of its leading principal minors are positive (Sylvester’s criterion).
Notes
More generally, a minor of order $k$ of a matrix, square or not, is the determinant of a $k \times k$ submatrix obtained by keeping $k$ of its rows and $k$ of its columns. In this sense, the $(i, j)$ minor of an $n \times n$ matrix is a minor of order $n - 1$. A principal minor keeps the same rows as columns, and a leading principal minor keeps the first $k$ rows and the first $k$ columns.