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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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="">5,>
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="">6,>
Therefore, the correct answer is option A.