MATH 1ZC3 Lecture Notes - Lecture 6: Row And Column Vectors, Coefficient Matrix, Invertible Matrix

• School: McMaster University
• Department: Mathematics
• Course: MATH 1ZC3
• Professor: Karsten Hempel

Document Summary

Cramer"s rule is an interesting by-product of invertibility and solving of systems. It is a way to calculate the solution to a system of linear equations, through only the calculation of determinants. Before we start, let"s mention that this method is not terribly efficient, especially for matrices of large dimension, but it is of significant theoretical importance, and becomes terribly important as a tool in solving differential equations in later years. So, although not the handiest process, it certainly has it"s import. You have seen previously systems of equations can be represented by: Where a is the matrix of coefficients, b, the column vector of constants, and x the column vector of the system"s variables. Now if a is invertible (and square) matrix, we can rewrite our expression using the inverse matrix, and it"s adjoint construction: If we examine adj a b more clearly, we see that something interesting happens: adj. C b nn n n i n i.