Definition:Submatrix

Let $$\mathbf{A}$$ be a matrix.

A submatrix of $$\mathbf{A}$$ is a matrix formed by selecting a subset of the rows and columns of $$\mathbf{A}$$, and using those entries (in the same relative positions) that appear in both the rows and columns of those selected.

For example, let $$\mathbf{A}$$ be as follows:



\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \end{bmatrix} $$

Then

\mathbf{A} \left[{1, 2; 1, 3, 4}\right] = \begin{bmatrix} a_{11} & a_{13} & a_{14} \\ a_{21} & a_{23} & a_{24} \end{bmatrix} $$

is a submatrix of $$\mathbf{A}$$ formed by rows $$1, 2$$ and columns $$1, 3, 4$$.

This submatrix can also be denoted by $$\mathbf{A} \left({3; 2}\right)$$ which means that it is formed by deleting row $$3$$ and column $$2$$.

The equivalent term for a determinant is a minor.