- #1
EvLer
- 458
- 0
Is the identity matrix (or multiple of) the only one that commutes with other matrices or are there other matrices that AB=BA?
Thanks
Thanks
An identity matrix is a square matrix with 1's down the main diagonal and 0's everywhere else. It is usually denoted by the symbol I and has the property that any matrix multiplied by the identity matrix will result in the original matrix.
The identity matrix serves as the multiplicative identity for matrices, much like how 1 serves as the multiplicative identity for real numbers. It also plays a crucial role in matrix operations and transformations.
No, the commutative property does not hold true for all matrices. In general, AB may not be equal to BA. However, for the identity matrix, AB = BA always holds true.
The proof for AB = BA for the identity matrix involves using the properties of matrix multiplication and the definition of the identity matrix. It can be shown that the multiplication of a matrix A with the identity matrix I results in the same matrix A, and similarly for B. Thus, AB=BA for the identity matrix.
Yes, there are other special matrices for which AB = BA holds true. These include the zero matrix, diagonal matrices, and scalar matrices. However, these are the only exceptions, and in general, AB may not be equal to BA for most matrices.