Classifying a 3×3 Linear System

00:00 4 marks

Consider the following 3×3 system of linear equations:

\(a_1 x + b_1 y + c_1 z = d_1\)
\(a_2 x + b_2 y + c_2 z = d_2\)
\(a_3 x + b_3 y + c_3 z = d_3\)

Describe conditions on the coefficients for which the system has a unique solution, infinitely many solutions, or no solution. Provide your answer in terms of relationships among the coefficients.

Hint Hint

Hint

Recall the concept of determinant for uniqueness and the relationships between the coefficient matrix and the augmented matrix for inconsistent/infinite solution scenarios.

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Answer Answer

Complete Answer

A unique solution exists if the determinant of the coefficient matrix is non-zero. Infinitely many solutions exist if the determinant is zero and the augmented matrix is consistent. No solution exists if the determinant is zero with an inconsistent augmented matrix.

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