A system of equations is said to be

consistentif there is at least one solution; it isinconsistentif there are no solutions.A linear system is inconsistent if and only if the reduced row-echelon form of its augmented matrix contains the row [ 0 0 ... 0 | 1], representing the equation 0 = 1.

If a linear system is consistent, then it has either

infinitely many solutions(if there is at least one free variable), orexactly one solution(if all the variables are leading).

The rank of a matrix A is the number of leading 1s in rref(A).

A linear system with fewer equations than unknowns has either no solutions or infinitely many solutions.

A linear system of n equations in n variables has a unique solution if and only if the rank of its coefficient matrix A is n.

- Sums of matrices
- Scalar multiples of matrices
- The identity matrix

- In terms of the columns of A
- In terms of the rows of A
- Algebraic rules

- Section 1.3 - 6, 9, 11, 19, 21, 29
- Chapter 1 Exercises - 3, 11, 15, 23, 35