Applications of Matrices and Determinants — Formula Sheet

AX=B

A=[[a1,b1,c1],[a2,b2,c2],[a3,b3,c3]], X=[x,y,z]^T, B=[d1,d2,d3]^T

If |A|≠0, then X=A^-1B

A^-1=(1/|A|)(adj A)

If |A|=0: inconsistent if (adj A)B≠O; consistent and dependent if (adj A)B=O

For A^T X=B, X=(A^T)^-1B=(A^-1)^T B

If AB=kI, then A^-1=(1/k)B

Δ=(1/2)|[x1 y1 1; x2 y2 1; x3 y3 1]|=(1/2)[x1(y2-y3)-y1(x2-x3)+(x2y3-x3y2)]

Area of triangle = |Δ|

Points A, B, C are collinear if Δ=0

Equation of line through (x1,y1) and (x2,y2): |[x y 1; x1 y1 1; x2 y2 1]|=0

The solution is given by X = A^{-1}B.

Use X = A^{-1}B only when the system has a unique solution.

For A^T X = B, use X = (A^{-1})^T B.