Linear Algebra Formulae

# Matrices and Systems

• Trace of $A_{n \times n}$: $\mathrm{tr}(A) = a_{11} + a_{22} + ... + a_{nn}$.
• Determinant of $A_{2 \times 2}$: $\mathrm{det}(A) = ad - bc$.
• Minor Entry in an $n \times n$ Matrix: $M_{ij} = \begin{vmatrix} a_{11} & a_{12} & ... & a_{1,i -1} & a_{1,i+1} & ... & a_{nn}\\ a_{21} & a_{22} & ... & a_{2, i-1} & a_{2,i+1} & ... & a_{2n}\\ \vdots & \vdots & & \vdots & \vdots & & \vdots\\ a_{i-1,1} & a_{i-1,2} & ... & a_{i-1,i-1} & a_{i-1,i+1} & ... & a_{i-1,n}\\a_{i+1,1} & a_{i+1,2} & ... & a_{i+1,i-1} & a_{i-1,i+1} & ... & a_{i+1,n}\\\vdots & \vdots & & \vdots & \vdots & & \vdots \\a_{n1} & a_{n2} & ... & a_{n,i-1} & a_{n,i+1} & ... & a_{nn}\end{vmatrix}$
• Cofactor Entry in an $n \times n$ Matrix: $C_{ij} = (-1)^{i + j} M_{ij}$