> [!summary]
A sub-matrix is a smaller matrix formed inside a larger matrix.
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# Definition
A sub-matrix is a smaller [[Matrix Notation|matrix]] $S$ formed by selecting certain rows and columns from a larger [[Matrix Notation|matrix]] $A$. The smaller matrix $S$ is a sub-matrix of $A$ if every [[Matrix Notation|entry]] in $S$ is also an entry in $A$.
# Examples
> [!example] Examples of a sub-matrix
> $
\text{Let } A =
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{bmatrix}
\quad \Rightarrow \quad
\text{Submatrix} =
\begin{bmatrix}
1 & 2 \\
7 & 8
\end{bmatrix}
> $
> $
\text{Let } B =
\begin{bmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12 \\
13 & 14 & 15 & 16
\end{bmatrix}
\quad \Rightarrow \quad
\text{Submatrix (rows 2-3, cols 2-4)} =
\begin{bmatrix}
6 & 7 & 8 \\
10 & 11 & 12
\end{bmatrix}
> $