GATE ME 2022 Set 2 — Question 46
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Engineering Mathematics → Linear Algebra → Rank of a Matrix
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Question
is a real matrix of rank 2. For the set of homogeneous equations , where is a zero vector and is a vector of unknown variables, which of the following is/are true?
Correct answer
(B) The given set of equations will be satisfied by a zero vector of appropriate size.; (C) The given set of equations will have infinitely many solutions.
Solution
For a homogeneous system , the zero vector is always a solution, so option (B) is correct. The number of variables is and the rank of the matrix is . The dimension of the solution space (nullity) is given by . Since the nullity is greater than zero, the system has infinitely many solutions. Therefore, option (C) is also correct.
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