GATE EE 2026 Set 1 — Question 56
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Engineering Mathematics → Linear Algebra → Existence & Uniqueness
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Question
Consider the system of linear equations: , where is an matrix, and and are -dimensional column vectors.Suppose this system of equations has a unique solution. Which of the following statements is/are correct?
Correct answer
(A) A⁻¹ exists; (B) The system of equations A^mx = b also has a unique solution for m = 1, 2, 3, …; (C) rank(A) = rank(A^m), for m = 1, 2, 3, …
Solution
A unique solution for an system implies that is non-singular (invertible).
(A) Since is non-singular, exists. Correct.
(B) If is non-singular, then is also non-singular for any positive integer . Thus, has a unique solution. Correct.
(C) . Since is also non-singular, . Correct.
(D) For a unique solution, . The statement says , which is incorrect.
(A) Since is non-singular, exists. Correct.
(B) If is non-singular, then is also non-singular for any positive integer . Thus, has a unique solution. Correct.
(C) . Since is also non-singular, . Correct.
(D) For a unique solution, . The statement says , which is incorrect.
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