Full-rank square matrix in RREF is the identity matrix

Dependencies:

  1. Reduced Row Echelon Form (RREF)
  2. Identity matrix

Let $A$ be an $n$ by $n$ matrix in RREF. Then all rows of $A$ are non-zero iff $A = I_n$ (identity matrix).

Proof

If $A$ is the identity matrix, all rows of $A$ are non-zero.

Let all rows of $A$ be non-zero. Let $\alpha_i$ be the smallest value of $j$ such that $A[i, j] \neq 0$. Since $A$ is in RREF, $\alpha_i < \alpha_{i+1}$. Therefore, $1 \le \alpha_1 < \alpha_2 < \ldots < \alpha_n \le n$. Therefore, $\alpha_i = i$.

Since $A$ is in RREF, $A[i, i] = A[i, \alpha_i] = 1$ and there is only one non-zero entry in the $\alpha_i^{\textrm{th}}$ column.

Therefore, $A$ is the identity matrix.

Dependency for:

  1. RREF([A|I]) = [I|inv(A)] iff A is invertible
  2. Full-rank square matrix is invertible
  3. A matrix is full-rank iff its determinant is non-0
  4. Determinant of product is product of determinants

Info:

Transitive dependencies:

  1. Group
  2. Ring
  3. Semiring
  4. Matrix
  5. Reduced Row Echelon Form (RREF)
  6. Identity matrix