GATE CS 2021 Set 2 — Question 34
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Engineering Mathematics → Linear Algebra → Rank of a Matrix
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Question
Suppose that is a matrix such that every solution of the equation is a scalar multiple of . The rank of is ___________.
Correct answer
4 to 4
Solution
The matrix is of size . The equation defines the null space (kernel) of .The problem states that every solution of is a scalar multiple of the non-zero vector . This implies that the null space of is spanned by a single vector, so the dimension of the null space (nullity) is .According to the Rank-Nullity Theorem:where is the number of columns of the matrix . Here, .
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