> [!summary] A sub-matrix is a smaller matrix formed inside a larger matrix. >[!info]+ Read Time **⏱ 1 min** # 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} > $