If a square matrix A is row equivalent to a matrix B, which of the following may

Question

If a square matrix A is row equivalent to a matrix B, which of the following may not be the same for both matrices?
A) Determinant B)Dimension of their null spaces C)Rank of matrices D)Number of solutions for homogeneous equations
Explanation would be extremely helpful if possible!
Thanks in advance If a square matrix A is row equivalent to a matrix B, which of the following may not be the same for both matrices?
A) Determinant B)Dimension of their null spaces C)Rank of matrices D)Number of solutions for homogeneous equations
Explanation would be extremely helpful if possible!
Thanks in advance
A) Determinant B)Dimension of their null spaces C)Rank of matrices D)Number of solutions for homogeneous equations
Explanation would be extremely helpful if possible!
Thanks in advance

Explanation / Answer

In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two m × n matrices are row equivalent if and only if they have the same row space.  Two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same null space.

So that means option B and option D is incorrect here as it is the must requirement here.

The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank.so option C is also incorrect.

So, only correct option is option a , THe determinant. If determinant can be different for the two matrices.

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