It’s not that I’m so smart. It’s just that I spent more time on problems.
– Albert Einstein (1879 - 1955)
Real-life matrices are often very large, so that many of the computations are performed using a computer. The purpose of this section is to explain a basic algorithm involved, namely row operations (also called row reduction or Gaussian elimination). This process works with matrices of any size and has many applications. These row operations are exactly the kind of steps one would take to solve a system of linear equations; we are simply going to encode these steps with symbols.
The idea is to modify a matrix, using a sequence of elementary steps. The modified matrix will let us easily understand properties of the original matrix. For instance, if a matrix encodes a system of linear equations, we modify it in order to obtain an equivalent system which is easier to solve and gives the same solution set as the original one.