2 Matrices

A matrix begins as a table of numbers, but the point of the section is that it is really an operation, and that the operation can be manipulated algebraically.

The section opens with the types of matrix and the rules of matrix arithmetic, which behave like ordinary arithmetic in every respect but one: multiplication does not commute. It then turns to elementary row operations, the systematic way of simplifying a matrix, and to the reduced echelon form they produce.

The result that makes everything afterwards possible is Theorem 2.6.3: each row operation is the same as multiplying on the left by a matrix. Once row reduction is matrix multiplication, it can be reasoned about rather than merely carried out, and that single fact is used to prove the theorems about systems in Section 3, about determinants in Section 4, and about rank in Section 7.

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