How to Determine Matrix Rank and Solve Linear Equation Systems

Theorem 1

Theorem 1: If the matrices are equivalent, then ...

Corollary: The rank of a matrix equals the number of non‑zero rows in its row‑echelon form.

According to the theorem and corollary, to find the rank of a matrix, transform it into row‑echelon form using elementary row operations; the number of non‑zero rows is the rank.

Homogeneous and Non‑Homogeneous Linear Systems

A linear system with m equations and n unknowns is generally written as ...

If the constant terms are not all zero, the system is called non‑homogeneous; otherwise it is homogeneous.

Theorem 2

Theorem 2: A homogeneous linear system has a non‑zero solution if and only if the rank R(A) of its coefficient matrix satisfies R(A) < n. This condition generalizes Cramer's rule, which only applies when m = n, and its sufficiency includes the converse of Cramer's rule.

Augmented Matrix

The solvability of a system depends on its coefficient matrix and the right‑hand vector, not on the notation of the unknowns. Therefore, we construct the augmented matrix, which combines the coefficient matrix and the right‑hand vector.

This matrix is called the augmented matrix of the system.

Theorem 3

Theorem 3: A non‑homogeneous linear system has a solution if and only if the rank of the coefficient matrix equals the rank of the augmented matrix.

Solving Linear Systems

The general steps to solve a linear system are:

Step 1: Apply elementary row operations to the augmented matrix to obtain its row‑echelon form, and compute the ranks of the coefficient matrix and the augmented matrix.

Step 2: If the ranks are different, the system has no solution; stop the computation. Otherwise, proceed to Step 3.

Step 3: Continue elementary row operations on the augmented matrix to reach reduced row‑echelon form. If the rank of the coefficient matrix equals the number of unknowns, write the unique solution of the system.

If r(A) = r(A|b) < n, the solution can be expressed in vector form.

Here, the vector represents the solution.