Understanding Invertibility and Pivots
In linear algebra, a square matrix A of size n × n is invertible if there exists a matrix A-1 such that AA-1 = I, where I is the n × n Identity matrix. But why does performing row operations on A always reduce it to I?
The answer lies in the fundamental properties of invertible matrices. By the Invertible Matrix Theorem, if a matrix is invertible, its columns must be linearly independent. Linear independence means that the only way to combine the columns to get the zero vector is to multiply each column by zero. In equation form, the system Ax = 0 has only the trivial solution, x = 0.
Row Operations Preserve the Null Space
An essential rule of matrix algebra is that elementary row operations do not change the solution set of Ax = 0 (the null space of the matrix). If the only solution to Ax = 0 is x = 0, then the only solution to RREF(A)x = 0 must also be x = 0.
Now let's look at the structure of RREF(A). If any column in RREF(A) did not contain a pivot, that column would correspond to a "free variable." A free variable allows us to find non-zero solutions to the system RREF(A)x = 0. Since we know no such solutions exist, every single column of RREF(A) must contain a pivot.
The Final RREF Structure
Because A is a square n × n matrix, and every column must contain a pivot, we must have exactly n pivots. By definition of RREF:
- Each pivot must be equal to 1.
- The pivot of each row must be to the right of the pivot in the row above it (forcing them onto the main diagonal).
- All other entries in a pivot's column must be 0.
A square matrix with 1s on the main diagonal and 0s everywhere else is, by definition, the Identity matrix. Thus, if a matrix is invertible, its RREF is guaranteed to be the Identity matrix.
The Contrapositive: What if RREF is Not the Identity?
If you perform row reduction on a square matrix and end up with even a single row of zeros (meaning RREF is not the Identity), the matrix is **singular** (not invertible). Its determinant is zero, its columns are linearly dependent, and it cannot be inverted.