Table of Contents
What can we say about A and B if both AB and BA are defined?
If AB and BA are defined, then AB and BA are square matrices.
How do you prove AB BA?
Need to show: AB = BA. Since A is not square, m = n. Therefore, the number of rows of AB is not equal to the number of rows of BA, and hence AB = BA, as required.
Do AB and BA have the same minimal polynomial?
Since A2 = 0 and A = 0, the minimal polynomial of AB is x2 whereas the minimal polynomial of BA is x. (2) Let A be an n × n matrix. So they have same characteristic polynomial and therefore same eigenvalues.
When can AB BA?
(19) can only be applied if we know that both A and B are invertible. In general, AB = BA, even if A and B are both square. If AB = BA, then we say that A and B commute. For a general matrix A, we cannot say that AB = AC yields B = C.
Does AB BA for matrices?
In general, AB = BA, even if A and B are both square. If AB = BA, then we say that A and B commute. For a general matrix A, we cannot say that AB = AC yields B = C. (However, if we know that A is invertible, then we can multiply both sides of the equation AB = AC to the left by A−1 and get B = C.)
What is the difference between AB and BA in matrix theory?
For example, if A = cI where I is the identity matrix, then AB = BA for all matrices B. In fact, the converse is true: If A is an n × n matrix such that AB = BA for all n × n matrices B, then A = cI for some constant c. Therefore, if A is not in the form of cI, there must be some matrix B such that AB ≠ BA. Share.
What is the eigenvalue of BA and AB with same eigenvalues?
So AB and BA are similar matrices, and they therefore have the same eigenvalues. (If x is an eigenvector of AB with eigenvalue \\lambda, then y=A^ {-1}x is the eigenvalue of BA with the same eigenvalue.) Tools for everyone who codes.
How to prove that $AB$ and $BA$ have the same characteristic polynomial?
The same goes if $B$ is invertible. In general, from the above observation, it is not too difficult to show that $AB$, and $BA$ have the same characteristic polynomial, the type of proof could depends on the field considered for the coefficient of your matrices though.
What is the product of A and B in the matrix?
Since A is 2 x 3 and B is 3 x 4, the product AB, in that order, is defined, and the size of the product matrix AB will be 2 x 4. The product BA is not defined, since the first factor (B) has 4 columns but the second factor (A) has only 2 rows.