Problem 71 - College Algebra and Calculus: An Applied Approach

Question

Reasoning Is it possible for a square linear system to have no solution? Explain.

Step-by-Step Solution

Answer

Yes, a square linear system can possibly have no solution. This happens when the determinant of the system's coefficient matrix is zero, leading to either no solution or an infinite number of solutions.

Step 1: Understand the meaning of the determinant

The determinant of a matrix is a particular computed number. This number tells if the matrix is invertible. If the determinant is zero, the system is either inconsistent (has no solution), or has an infinite number of solutions.

Step 2: Recognize the nature of square linear system

A square linear system is one in which the number of equations equals the number of unknowns. In other words, if we have 'n' equations for 'n' unknowns, then it is a square system.

Step 3: Connect square linear system and determinant

We compute the determinant of the coefficient matrix for a square linear system. If the result is non-zero, it indicates that the system has a unique solution. If it's zero, however, the system either has no solution or has an infinite number of solutions. In this sense, it is indeed possible for a square linear system to have no solution.

Key Concepts

Determinant

The determinant is an essential concept in linear algebra, particularly when dealing with square linear systems. A matrix's determinant provides a numeric value that helps identify certain characteristics of the matrix. For a square matrix, the determinant can be calculated from its elements in a way that involves adding and subtracting products of its entries. This number is crucial because it tells us several things about the matrix:

• If the determinant is non-zero, the matrix is invertible, meaning solutions to related systems are possible and unique.
• If the determinant is zero, it indicates that the matrix is not invertible, often leading to special cases for solutions.

Knowing how to compute and interpret the determinant is key in understanding the general nature of linear systems and matrix algebra.

Matrix Invertibility

Matrix invertibility is closely tied to the concept of the determinant. In the realm of linear algebra, a matrix is said to be invertible, or non-singular, if there exists another matrix that, when multiplied with the original, yields the identity matrix. This situation can only arise when the determinant of the matrix is non-zero.
When a matrix is invertible, a unique solution exists for the associated linear system, assuming it is consistent. This process is akin to division in arithmetic. If the matrix is not invertible, the solutions to the system are either nonexistent or not unique. Hence, invertibility is a central idea when discussing whether solutions to matrix equations exist or are unique.

Unique Solution

A unique solution is an outcome where a linear system of equations has precisely one set of values for the unknowns that satisfies all equations simultaneously. In the context of a square linear system, which has the same number of equations as variables, a unique solution occurs if and only if the determinant of the coefficient matrix is non-zero.

• This non-zero determinant ensures matrix invertibility, allowing for the resolution of the system.
• Such a system is said to be consistent and uniquely determined, meaning there is no ambiguity in the solution.

In practical terms, having a unique solution means every variable in the system has a single, definitive value that satisfies all given equations, highlighting the power of the determinant in predicting solution behavior.

Inconsistent System

An inconsistent system is one where no solution exists. For a square linear system, this inconsistency is often indicated when the determinant of the coefficient matrix is zero. In such cases, either no solutions or infinitely many solutions exist, with inconsistency referring specifically to the scenario of no feasible solutions that satisfy the system.
An inconsistent system can arise from the equations contradicting each other. For example, if two equations in a system represent parallel lines, they will never intersect, leading to no common solution. The zero determinant hints at this possibility, emphasizing that even with a square system, solutions aren't guaranteed unless specific mathematical conditions are met, such as non-zero determinants.