ImA, or the image of matrix A, is the set of all possible outputs or column vectors that can be obtained by multiplying A with any input or column vector.
In, or the identity matrix, is a square matrix with 1s along the main diagonal and 0s everywhere else. Multiplying any matrix with In will result in the original matrix, similar to multiplying any number with 1.
If ImA = A, it means that all the outputs or column vectors that can be obtained by multiplying A with any input or column vector are identical to the original matrix A. In other words, the image of A is the same as A itself.
To prove that ImA = A, we can show that every column vector in A is a linear combination of the columns in A. This can be done by row-reducing A and observing that the pivot columns correspond to the original columns in A.
The equation AIn = A indicates that multiplying A with the identity matrix In does not change the matrix A. This is a useful property in matrix operations and can be used to simplify calculations.