GATE Chemical 2019 | Question: 1

Question

A system of $n$ homogeneous linear equations containing $n$ unknowns will have non-trivial solutions if and only if the determinant of the coefficient matrix is

• A. $1$
• B. $-1$
• C. $0$
• D. $\infty$

Answer 1

• Understanding Homogeneous Systems: A system is homogeneous if it is of the form $AX = 0$. This system always has at least the "trivial" solution where $X = 0$.

• Condition for Non-Trivial Solutions: A system of $n$ equations with $n$ unknowns has non-trivial solutions if and only if the matrix $A$ is singular, meaning it is not invertible.

• Determinant Property: A matrix is singular if and only if its determinant is exactly 0.

Final Answer: Option C