GATE EC 2025 Set 1 — Question 11
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Engineering Mathematics → Linear Algebra → Rank of a Matrix
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Question
Consider the matrix below:For which of the following combinations of , and , is the rank of at least three?(i) and .
(ii) .
(iii) and .
(iv) .
(ii) .
(iii) and .
(iv) .
Correct answer
(A) Only (i), (iii), and (iv)
Solution
The matrix is in upper triangular form. The rank of an upper triangular matrix is the number of non-zero rows. Rows 1 and 2 are clearly non-zero. Thus, the rank is at least 3 if at least one of Row 3 or Row 4 is non-zero.Row 3 is non-zero if or .
Row 4 is non-zero if .Evaluating the cases:
(i) : Row 3 is and Row 4 is . Since , both rows are non-zero. However, they are linearly dependent. The rank is 3. (Satisfies rank )
(ii) : Rows 3 and 4 are zero rows. Rank is 2. (Does not satisfy)
(iii) : Row 3 is and Row 4 is zero. Rank is 3. (Satisfies rank )
(iv) : Row 3 is and Row 4 is . Both are non-zero and linearly independent. Rank is 4. (Satisfies rank )Therefore, combinations (i), (iii), and (iv) result in a rank of at least three.
Row 4 is non-zero if .Evaluating the cases:
(i) : Row 3 is and Row 4 is . Since , both rows are non-zero. However, they are linearly dependent. The rank is 3. (Satisfies rank )
(ii) : Rows 3 and 4 are zero rows. Rank is 2. (Does not satisfy)
(iii) : Row 3 is and Row 4 is zero. Rank is 3. (Satisfies rank )
(iv) : Row 3 is and Row 4 is . Both are non-zero and linearly independent. Rank is 4. (Satisfies rank )Therefore, combinations (i), (iii), and (iv) result in a rank of at least three.
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