GATE 2015 EE (EE2) – Question 12

Engineering Mathematics · Linear Algebra: Systems of linear equations · 1 mark · Multiple choice

For three linear equations in three unknowns, define P: unique solution; Q: independent equations; R: all coefficient-matrix eigenvalues nonzero; S: determinant nonzero. Here ≡ means equivalent and ≢ means not equivalent. Which is true?

$P\equiv Q\equiv R\equiv S$
$P\equiv R\not\equiv Q\equiv S$
$P\equiv Q\not\equiv R\equiv S$
$P\not\equiv Q\not\equiv R\not\equiv S$

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Correct answer: (A) $P\equiv Q\equiv R\equiv S$

Explanation

For a square coefficient matrix, a unique solution is equivalent to invertibility. Invertibility is equivalent to independent coefficient rows and to nonzero determinant.

The determinant is the product of eigenvalues, so it is nonzero exactly when all eigenvalues are nonzero. Under the usual meaning of independent coefficient equations, all four statements are equivalent: **A**.

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