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## Precalculus

### Course: Precalculus>Unit 7

Lesson 11: Properties of matrix multiplication

# Dimensions of identity matrix

Sal explains why the identity matrix is always a square matrix, even though it works with non-square matrices. Created by Sal Khan.

## Want to join the conversation?

• I dont understand. I thought IC=CI. But if IC=CI then in this case CI is not defined. Because the number of columns of C is not the same as the number of rows of I. •  IC = C
CI = C
If C is a square matrix, then the dimensions of the identity matrices in IC and CI are the same.
If C isn't a square matrix, then the dimensions of the identity matrices in IC and CI will be different but both equations will still be valid.
• Just curious, is the identity matrix always going to be a square matricies? I'm not sure how it would work if it wasn't a square matricies (if that's possible). Thanks. • What is a 1x1 identity matrix? • Is the Identity Matrix commutative? • Just to make it clear when i get a MCQ on this one. So the commutative property does work in matrices but not in all cases. right?? • The commutative property does not work with matrices. This is obvious with matrices that have dissimilar dimensions, but it is also true for square, or nxn matrices.

Matrices are part of a "family", (we use the description "algebraic ring" in math) that have the particular property that they are non commutative (most rings are commutative, such as the real numbers, which is also are a ring).

What it means to be a non-commutative ring is that there exists an a and b such that a·b≠b·a.
It is a rare case that a·b=b·a for any a or b that is not the zero matrix nor the identity matrix.

So, the only time that commutivity is assured with matrices is when at least one of a or b is the zero matrix or is the identity matrix.

The bottom line regarding commutivity and matrices is exactly the opposite of what you stated.
The commutative property DOES NOT work with matrices except in the case of the identity and/or zero matrix
• How can we apply identity matrix? Is it designed to simplify any process or calculation? Thanks. • What would be the dimensions of the product CxI, given that dimensions of C are axb in which a and b are different values ? In this case would the dimensions of I be bxb ? • If matrix A is non-squared, then the identity matrix in the equation IC = C is different from the identity matrix in CI = C... correct? But ... then CI DOES NOT equal IC?? Because you changed the I? Similarly, AB does not equal BA, if you suddenly say that the B is now something different than before? So am I missing something, or is IC = CI only valid for squared matrices? • Yes, the left and right identity matrices are not the same size, and are therefore not the same matrix. Let's say I_l (capital i underscore lowercase L) is the left identity matrix, and that I_r is the right identity matrix.

I_l A = A I_r is valid for all matrices, since by definition, I_l A = A and A I_r = A. The fact that matrix multiplication isn't commutative is not relevant here.
(1 vote)
• 3b1b's essense of Linear Algebra was really superb
I understood a lot of the optics in matrices including these in a really intuitive way.
The reason the I matrix is always square is because (in this case, where I is on the left), it is used as a transformation matrix.

The matrix on the right can be treated as vector with n dimensions where n is the number of rows on that 2nd matrix. The I matrix has to match the number the number of dimensions of that 2nd matrix then by matching the number of rows through its columns.

The reason they're square though is because they are BASIS VECTORS where none of the vectors are linearly dependent. In order for that to happen, they must live in different dimensions. So the number of dimensions of that I matrix is the same as its number of columns.
But we already know that number of col = num of rows of the 2nd matrix. Therefore the I matrix would be n*n where n=num of of of the 2nd matrix.  