Lecture 3: Inverse of Matrix and Gauss - Jordan Elimination Method

Inverse Matrix:
Let A be a square n by n matrix over a field K (for example the field R of real numbers). The following statements are equivalent, i.e., for any given matrix they are either all true or all false:

A is invertible, i.e. A has an inverse, is nonsingular, or is nondegenerate.
A is row-equivalent to the n-by-n identity matrix In.
A is column-equivalent to the n-by-n identity matrix In.
A has n pivot positions.
det A ? 0. In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring.
A has full rank; that is, rank A = n.
The equation Ax = 0 has only the trivial solution x = 0.
Null A = {0}.

Gauss-Jordan Elimination Method
The following row operations on the augmented matrix of a system produce the augmented matrix
of an equivalent system, i.e., a system with the same solution as the original one.
• Interchange any two rows.
• Multiply each element of a row by a nonzero constant.
• Replace a row by the sum of itself and a constant multiple of another row of the matrix.
For these row operations, we will use the following notations.
• Ri ? Rj means: Interchange row i and row j.
• ?Ri means: Replace row i with ? times row i.
• Ri + ?Rj means: Replace row i with the sum of row i and ? times row j.
The Gauss-Jordan elimination method to solve a system of linear equations is described in the
following steps.
1. Write the augmented matrix of the system.
2. Use row operations to transform the augmented matrix in the form described below, which is
called the reduced row echelon form (RREF).
(a) The rows (if any) consisting entirely of zeros are grouped together at the bottom of the
matrix.
(b) In each row that does not consist entirely of zeros, the leftmost nonzero element is a 1
(called a leading 1 or a pivot).
(c) Each column that contains a leading 1 has zeros in all other entries.
(d) The leading 1 in any row is to the left of any leading 1’s in the rows below it.
3. Stop process in step 2 if you obtain a row whose elements are all zeros except the last one on
the right. In that case, the system is inconsistent and has no solutions. Otherwise, finish step
2 and read the solutions of the system from the final matrix.