The inverse of a matrix
It would be nice to have a way to do algebra on matrices the way we do for simple algebraic expressions, like so:
The inverse of a matrix provides us with a way to do this. It is very similar to the reciprocal for rational numbers. For rational numbers, the following is true:
In a similar way, the inverse of a matrix is defined to be a matrix that when multiplied by the original matrix, you get the identity matrix. Here it is mathematically:
The matrix inverse can then be used when trying to algebraically modify a matrix equation. Let's say we are trying to find the vector |x⟩ in the following equation:
Since we now have a multiplicative inverse of a matrix, we can multiply both sides by it to get the following:
Please remember that matrix multiplication is not commutative, so if you left multiply a matrix on one...
