Inside Collection: Matrices
Summary: This module introduces the identity matrix and its properties.
When multiplying numbers, the number 1 has a special property: when you multiply 1 by any number, you get that same number back. We can express this property as an algebraic generalization:
The matrix that has this property is referred to as the identity matrix.
The identity matrix, designated as
Note that the definition of [I] stipulates that the multiplication must commute—that is, it must yield the same answer no matter which order you multiply in. This is important because, for most matrices, multiplication does not commute.
What matrix has this property? Your first guess might be a matrix full of 1s, but that doesn’t work:
so

The matrix that does work is a diagonal stretch of 1s, with all other elements being 0.
so



You should confirm those multiplications for yourself, and also confirm that they work in reverse order (as the definition requires).
Hence, we are led from the definition to:
For any square matrix, its identity matrix is a diagonal stretch of 1s going from the upperlefthand corner to the lowerright, with all other elements being 0.
Nonsquare matrices do not have an identity. That is, for a nonsquare matrix
Why no identity for a nonsquare matrix? Because of the requirement of commutativity. For a nonsquare matrix
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