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Manipulating Matrix Equations

If we have a matrixthat satisfies the equationwhereis the identity matrix then we can findby multiplying throughout by

so the above equation becomes

ando

We can rearrange this to makethe subject.

The process is a lot like ordinary algebra, with the exception that when using matrices we must be careul to multiply byon the left throughout or on the right throughour, since matrix multiplication is not commutative. This does not matter if onlyand the identity matrix appear in the equation, but it is still a good habit to acquire.

Example: Ifthen

so that multiplying throughout bygivesand