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If the rank of a (5 x 6) matrix Q is 4, then which one of the following statements is correct?
  • a)
    Q will have four linearly independent rows and four linearly independent columns
  • b)
    Q will have four linearly independent rows and five linearly independent columns
  • c)
    QQT will be invertible
  • d)
    QTQ will be invertible
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If the rank of a (5 x 6) matrix Q is 4, then which one of the followin...
If rank of (5 x 6) matrix is 4, then surely it must have exactly 4 linearly independent rows as will as 4 linearly in dependent columns.
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Most Upvoted Answer
If the rank of a (5 x 6) matrix Q is 4, then which one of the followin...
Explanation:
Rank of a matrix is the maximum number of linearly independent rows or columns in a matrix. So, if the rank of a (5 x 6) matrix Q is 4, it means that there are 4 linearly independent rows or columns.

Option A: Q will have four linearly independent rows and four linearly independent columns.
If Q has 4 linearly independent rows, it means that there are 5 - 4 = 1 dependent row. Similarly, if Q has 4 linearly independent columns, it means that there are 6 - 4 = 2 dependent columns. Therefore, Q will have 1 dependent row and 2 dependent columns. Hence, option A is correct.

Option B: Q will have four linearly independent rows and five linearly independent columns.
If Q has 4 linearly independent rows, it means that there are 5 - 4 = 1 dependent row. But if Q has 5 linearly independent columns, it means that there are 6 - 5 = 1 dependent column. Therefore, Q will have 1 dependent row and 1 dependent column. Hence, option B is incorrect.

Option C: QQT will be invertible
QQT is a (5 x 5) matrix. If Q has rank 4, then QQT will also have rank 4 or less. An invertible matrix has full rank, i.e., rank=n (where n is the number of rows or columns). Therefore, if rank(QQT)<5, then="" qqt="" will="" not="" be="" invertible.="" hence,="" option="" c="" is="">

Option D: QTQ will be invertible
QTQ is a (6 x 6) matrix. If Q has rank 4, then QTQ will also have rank 4 or less. An invertible matrix has full rank, i.e., rank=n (where n is the number of rows or columns). Therefore, if rank(QTQ)<6, then="" qtq="" will="" not="" be="" invertible.="" hence,="" option="" d="" is="">

Therefore, the correct answer is option A.
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If the rank of a (5 x 6) matrix Q is 4, then which one of the following statements is correct?a)Q will have four linearly independent rows and four linearly independent columnsb)Q will have four linearly independent rows and five linearly independent columnsc)QQT will be invertibled)QTQ will be invertibleCorrect answer is option 'A'. Can you explain this answer?
Question Description
If the rank of a (5 x 6) matrix Q is 4, then which one of the following statements is correct?a)Q will have four linearly independent rows and four linearly independent columnsb)Q will have four linearly independent rows and five linearly independent columnsc)QQT will be invertibled)QTQ will be invertibleCorrect answer is option 'A'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about If the rank of a (5 x 6) matrix Q is 4, then which one of the following statements is correct?a)Q will have four linearly independent rows and four linearly independent columnsb)Q will have four linearly independent rows and five linearly independent columnsc)QQT will be invertibled)QTQ will be invertibleCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the rank of a (5 x 6) matrix Q is 4, then which one of the following statements is correct?a)Q will have four linearly independent rows and four linearly independent columnsb)Q will have four linearly independent rows and five linearly independent columnsc)QQT will be invertibled)QTQ will be invertibleCorrect answer is option 'A'. Can you explain this answer?.
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