Proof Prove each property of the system of linear equations...

Proof Prove each property of the system of linear equations in $n$ variables $A \mathbf{x}=\mathbf{b}$ (a) If rank(A) $=\operatorname{rank}([A, \mathbf{b}])=n,$ then the system has a unique solution. (b) If rank(A) $=\operatorname{rank}([A \quad \mathbf{b}])<n,$ then the system has infinitely many solutions. (c) If $\operatorname{rank}(A)<\operatorname{rank}([A \quad b]),$ then the system is inconsistent.

Proof Prove each property of the system of linear equations in $n$ variables $A \mathbf{x}=\mathbf{b}$
(a) If rank(A) $=\operatorname{rank}([A, \mathbf{b}])=n,$ then the system has
a unique solution.
(b) If rank(A) $=\operatorname{rank}([A \quad \mathbf{b}])<n,$ then the system has infinitely many solutions.
(c) If $\operatorname{rank}(A)<\operatorname{rank}([A \quad b]),$ then the system is
inconsistent.

Key Concepts

Matrix Rank

Matrix rank is a fundamental concept in linear algebra that measures the number of linearly independent rows or columns in a matrix. It essentially tells us the maximum number of independent equations in a system, and hence plays a crucial role in determining whether a solution exists and if it is unique.

Augmented Matrix

The augmented matrix is formed by appending the column vector of constants to the coefficient matrix of the system. This construct is used to analyze the system’s consistency by comparing the rank of the coefficient matrix with that of the augmented matrix.

Rouché–Capelli Theorem

The Rouché–Capelli theorem provides a clear criterion for the existence of solutions in a system of linear equations. It states that a system is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix, and it distinguishes cases where the solution is unique (when the common rank equals the number of variables) or infinite (when the rank is less than the number of variables).

Parametric Solutions and Free Variables

When the rank of the matrix is less than the number of unknowns, there exist free variables that can take on arbitrary values. These free variables lead to a family of solutions described parametrically, meaning the general solution can be expressed in terms of these parameters, resulting in infinitely many solutions.

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