GATE EE 2015 Set 2 — Question 12
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Engineering Mathematics → Linear Algebra → Determinants
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Question
We have a set of 3 linear equations in 3 unknowns. '' means and are equivalent statements and '' means and are not equivalent statements.P: There is a unique solution.
Q: The equations are linearly independent.
R: All eigenvalues of the coefficient matrix are nonzero.
S: The determinant of the coefficient matrix is nonzero.Which one of the following is TRUE?
Q: The equations are linearly independent.
R: All eigenvalues of the coefficient matrix are nonzero.
S: The determinant of the coefficient matrix is nonzero.Which one of the following is TRUE?
Correct answer
(A) P ≡ Q ≡ R ≡ S
Solution
For a system of linear equations in unknowns :
- P: A unique solution exists if and only if is non-singular ().
- Q: The equations are linearly independent if and only if the rows of are linearly independent, which means is non-singular.
- R: All eigenvalues of are nonzero if and only if their product (the determinant) is nonzero, which means is non-singular.
- S: The determinant of is nonzero means is non-singular.
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