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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

darrrrUC
  • 331

2 Answers2

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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

$$X+2I=B+XA\\ X-XA=B-2I\\ X(I-A)=B-2I\\ X=(B-2I)(I-A)^{-1}$$

Wouter
  • 7,673
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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 :

$$X = (B - 2I)(I - A)^{-1}$$

brick
  • 1,919