Question

If A is invertible matrix and B is any matrix, then
A. \[Rank(AB) = Rank(A)\]
B. \[Rank(AB) = Rank(B)\]
C. \[Rank(AB) > Rank(A)\]
D. \[Rank(AB) > Rank(B)\]

Answer 1

Hint: Rank of a matrix is the maximum number of linearly independent rows in the matrix. To find rank of a matrix we perform elementary row or column operations to convert it into Echelon form, then, the number of non-zero rows in the matrix gives us the rank of the matrix.
* Product of two matrices exists if and only if number of columns of first matrix is equal to
   number of rows of the second matrix.
* For a matrix \[A\] of order \[m \times n\], \[{A_{m \times n}} \leqslant \min (m,n)\].
* Invertible matrices are the square matrix whose inverse exists.
* If product of two matrices exist, then rank of product of the matrix will not be more than the rank of either matrix, that is \[{\text{Rank}}\left( {AB} \right) \leqslant {\text{Rank}}\left( A \right)\] and \[{\text{Rank}}\left( {AB} \right) \leqslant {\text{Rank}}\left( B \right)\]

Complete step by step solution:
Given, \[A\] is an invertible matrix, that is its inverse is defined.
Matrix \[B\] can be written as \[B = {A^{ - 1}}\left( {AB} \right)\] provided product \[AB\]is defined and exists.
Therefore, \[{\text{Rank}}\left( B \right) = {\text{Rank}}\left( {{A^{ - 1}}\left( {AB} \right)} \right)\].
From the property of rank of product of matrices, Rank of \[B\] cannot be greater than Rank of either \[{A^{ - 1}}\] or \[AB\], that is \[{\text{Rank}}\left( B \right) \leqslant {\text{Rank}}\left( {AB} \right)\,\,\,\,\, \ldots \left( 1 \right)\].
When \[AB\] is considered, which is the product of matrix \[A\] and \[B\]
i.e. \[AB = A \times B\]
From the property of rank of product of matrices, Rank of \[AB\] cannot be greater than Rank of either \[A\] or \[B\],
Therefore, \[{\text{Rank}}\left( {AB} \right)\, \leqslant \,{\text{Rank}}\left( B \right)\,\, \ldots \left( 2 \right)\].
From equations \[(1)\] and \[(2)\] , it can be concluded that \[{\text{Rank}}\left( {AB} \right){\text{ = Rank}}\left( B \right)\].

Therefore, option B is correct.

Note: In these types of questions where rank of product of matrix is to be determined, the product of the matrix should be defined, otherwise the rank doesn’t exist. Students should focus on comparing the ranks on the basis of the fact that rank of product of two matrices will always be less than or equal to either of the matrices.