Matrices

At a glance 10 formulas

1. 1Order of a product$(m \times n)(n \times p) \to m \times p$$AB \text{ needs columns of } A = \text{rows of } B$
2. 2Entry of a product$(AB)_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots$$\text{row } i \text{ of } A \text{ times column } j \text{ of } B$
3. 3Order matters$AB \ne BA \text{ in general}$$(A + B)^2 = A^2 + AB + BA + B^2$
4. 4Transpose of a product$(AB)^T = B^TA^T$$A^T \text{ turns rows into columns};\ (A^T)^T = A$
5. 5Symmetric and skew$A^T = A \text{ (symmetric)},\ A^T = -A \text{ (skew-symmetric)}$$\text{skew-symmetric: every diagonal entry is } 0$
6. 6Inverse of a 2×2$\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \dfrac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$\text{none if } ad - bc = 0;\ (AB)^{-1} = B^{-1}A^{-1}$
7. 7Cayley-Hamilton (2×2)$A^2 = (a + d)A - (ad - bc)I$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix},\ I = \text{identity matrix}$
8. 8Special squares$A^2 = I \Rightarrow A^{\text{even}} = I,\ A^{\text{odd}} = A$$A^2 = A \Rightarrow A^n = A \text{ for every } n;\ I = \text{identity}$
9. 9Square is zero$A^2 = O \Rightarrow (I + A)^n = I + nA$$O = \text{zero matrix},\ I = \text{identity}$
10. 10Entry sum of AB$\text{sum of entries of } AB = c_1r_1 + c_2r_2 + \cdots$$c_k = \text{column } k \text{ sum of } A,\ r_k = \text{row } k \text{ sum of } B$