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### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot position in each column. Explain why the system has a unique solution.

The system has a unique solution because it is row reduced to an identity matrix of the order $$3 \times 3$$.

See the step by step solution

## Step 1: Identify the condition for a unique solution

The system of three equations is fulfilled if the three planes cross at a single location.

As a result, the echelon form that can be produced by solving this system of equations is an identity matrix of the order $$3 \times 3$$.

Thus, the echelon form of the consistent linear system is an identity matrix of the order $$3 \times 3$$.

## Step 2: Prove that the system has a unique solution

It is given that the coefficient matrix of a linear system of three equations in three variables has a pivot position in each column. So, the system in span $${\mathbb{R}^3}$$ is row reduced to identity matrix echelon form.

Thus, the system has a unique solution.