I have two Know matrices A and B and I am given the equation:
X + 2I (identity) = B + XA
What are the operations allowed here? I wish to get something like X = ... So I can calculate the known matrices and get my X. Anywhere I can find a tutorial on this type of operation? Im not sure what its called or how to perform them correctly.
Cheers
On
You can do everything you can do with scalars (numbers) BUT take into account that matrix multiplication is not in general commutative $AB\neq BA$ and if you want to divide you have to explicitly multiply by the inverse $A X = B \implies X = A^{-1} B$. Addition and subtraction behave as they do on numbers. Multiplication with the identity does always commute $X I = I X=X$, as does multiplication with 0: $X 0 = 0 X = 0$.
Matrices (with elements in the real numbers) form a ring.
So
$$X+2I=B+XA\\ X-XA=B-2I\\ X(I-A)=B-2I\\ X=(B-2I)(I-A)^{-1}$$
You can use the operations $+, -$ (componentwise over the elements of the matrices, assuming they have the same dimensions) multiplication with scalars (again componentwise) and the regular multiplication of matrices.
$$X + 2I = B + XA \Leftrightarrow XI + 2I = B + XA \Leftrightarrow XI - XA = B - 2I \Leftrightarrow X(I - A) = B - 2I$$
If the the matrix $(I - A)$ is invertible (i.e. its determinant is not $0$) then :
$$X = (B - 2I)(I - A)^{-1}$$