Assuming your work so far is correct (I didn't check), set the elements of AB equal to those of BA. This will give you four equations in your four variables.leo255 said:Homework Statement
[/B]
Matrix A =
1 1
0 1
Matrix B =
a b
c d
Find coordinates on a, b, c, d such that AB = BA.
Homework Equations
The Attempt at a Solution
I calculated AB and BA with simple matrix multiplication, but am not sure where to go from here.
AB =
a + c a + b
c | d
BA =
a | a + b
c | c + d
To find the coordinates a, b, c, d for the matrix AB=BA, you need to use the properties of linear algebra. Since AB=BA, we know that these matrices are equal, so you can set up a system of equations using the entries of A and B. Then, you can solve for a, b, c, d by using Gaussian elimination or another method of solving systems of equations.
Yes, the equality AB=BA in linear algebra has several important implications. It shows that the matrices A and B commute, meaning that the order in which they are multiplied does not matter. This also means that A and B share the same eigenvectors, which can be useful in solving certain problems.
In order for AB=BA to be possible, the matrices A and B must be square matrices of the same size. This means that they have the same number of rows and columns. Additionally, the matrices must have compatible dimensions, meaning that the number of columns in A must be equal to the number of rows in B.
One real-world application of finding coordinates a, b, c, d for AB=BA is in computer graphics. Matrices are commonly used to represent transformations in computer graphics, and being able to solve for the coordinates allows for efficient and accurate rendering of 3D objects. This concept is also important in quantum mechanics and other areas of physics.
There are some shortcuts or tricks that can be used to solve for coordinates a, b, c, d in AB=BA, such as using the properties of matrices to simplify the equations or using matrices with special patterns, such as diagonal or triangular matrices. However, in general, the most efficient and accurate method is to use Gaussian elimination or another method of solving systems of equations.