Block Matrix

Definition

A matrix that is interpreted as having been broken into sections called blocks or sub-matrices

Operations

Addition, Subtraction

$\left[\begin{array}{ccc}{\mathbf{A}}_{11}& \dots & {\mathbf{A}}_{1n}\\ ⋮& & ⋮\\ {\mathbf{A}}_{m1}& \dots & {\mathbf{A}}_{mn}\end{array}\right]\pm \left[\begin{array}{ccc}{\mathbf{B}}_{11}& \dots & {\mathbf{B}}_{1n}\\ ⋮& & ⋮\\ {\mathbf{B}}_{m1}& \dots & {\mathbf{B}}_{mn}\end{array}\right]=\left[\begin{array}{ccc}{\mathbf{A}}_{11}\pm {\mathbf{B}}_{11}& \dots & {\mathbf{A}}_{1n}\pm {\mathbf{B}}_{1n}\\ ⋮& & ⋮\\ {\mathbf{A}}_{m1}\pm {\mathbf{B}}_{m1}& \dots & {\mathbf{A}}_{mn}\pm {\mathbf{B}}_{mn}\end{array}\right]$

Scalar Multiplication

$c\left[\begin{array}{ccc}{\mathbf{A}}_{11}& \dots & {\mathbf{A}}_{1n}\\ ⋮& & ⋮\\ {\mathbf{A}}_{m1}& \dots & {\mathbf{A}}_{mn}\end{array}\right]=\left[\begin{array}{ccc}c{\mathbf{A}}_{11}& \dots & c{\mathbf{A}}_{1n}\\ ⋮& & ⋮\\ c{\mathbf{A}}_{m1}& \dots & c{\mathbf{A}}_{mn}\end{array}\right]$

Matrix Multiplication

$\left[\begin{array}{ccc}{\mathbf{B}}_{11}& \dots & {\mathbf{B}}_{1n}\\ ⋮& & ⋮\\ {\mathbf{B}}_{k1}& \dots & {\mathbf{B}}_{kn}\end{array}\right]\left[\begin{array}{ccc}{\mathbf{A}}_{11}& \dots & {\mathbf{A}}_{1n}\\ ⋮& & ⋮\\ {\mathbf{A}}_{m1}& \dots & {\mathbf{A}}_{mn}\end{array}\right]=\left[\begin{array}{ccc}({\mathbf{B}}_{11}{\mathbf{A}}_{11}+\cdots +{\mathbf{B}}_{1m}{\mathbf{A}}_{m1})& \dots & ({\mathbf{B}}_{11}{\mathbf{A}}_{1n}+\cdots +{\mathbf{B}}_{1m}{\mathbf{A}}_{mn})\\ ⋮& & ⋮\\ ({\mathbf{B}}_{k1}{\mathbf{A}}_{11}+\cdots +{\mathbf{B}}_{km}{\mathbf{A}}_{m1})& \dots & ({\mathbf{B}}_{k1}{\mathbf{A}}_{1n}+\cdots +{\mathbf{B}}_{km}{\mathbf{A}}_{mn})\end{array}\right]$

Determinant

Let $D$ or $A$ be invertible matrices $\left|\begin{array}{cc}A& B\\ C& D\end{array}\right|=|D|\cdot |A-B{D}^{-1}C|=|A|\cdot |D-C{A}^{-1}B|$