If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

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?

  • female robot grace style avatar for user sunnymunich1158
    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.
    (17 votes)
    Default Khan Academy avatar avatar for user
    • aqualine tree style avatar for user snowjay99
      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.
      (43 votes)
  • leafers ultimate style avatar for user Tim Wu
    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.
    (15 votes)
    Default Khan Academy avatar avatar for user
  • male robot johnny style avatar for user Neil Muglurmath
    What is a 1x1 identity matrix?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Al V.
    Is the Identity Matrix commutative?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • duskpin ultimate style avatar for user Karthikeyan SARAVANABAVAN
    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??
    (3 votes)
    Default Khan Academy avatar avatar for user
    • leaf blue style avatar for user Stefen
      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
      (4 votes)
  • aqualine ultimate style avatar for user Patrick Zhou Zitong
    How can we apply identity matrix? Is it designed to simplify any process or calculation? Thanks.
    (4 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user Henry Li
      Its just good to know what 1 is and really get use to the properties of matrices. Using this basic concept you can know how to adjust numbers to make other matrix you encounter easily to comprehend and summarize. As well as you can practice some essential steps such as rows and column rules.
      (1 vote)
  • duskpin ultimate style avatar for user tuannb1997
    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 ?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • aqualine tree style avatar for user White
    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.
    (2 votes)
    Default Khan Academy avatar avatar for user
  • starky tree style avatar for user Slotynou
    Do we agree that, in this example, if we take CI=C then I would be a bxb square matrix ?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • leaf green style avatar for user Baber Javaid
    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?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user ArDeeJ
      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)

Video transcript

Voiceover:Let's say that you've got some matrix C, trying my best to bold it, to make sure you realize that this is a matrix. Let's say that we know that it has a rows and b columns. It's an a by b matrix. Let's say that we are going to multiply it by some identity matrix. Once again let me do my best to attempt to bold this right over here. We're going to multiply the identity matrix I times C and of course we are going to get C again because that's the identity matrix, that's the property of the identity matrix. Of course C, we already know is an a by b matrix. A rows and b columns. Based on this, what are the dimensions of I going to be? I encourage you to pause this video and think about it on your own. We've already done this exercise a little bit, where we first looked at identity matrices but now we're doing it with a very ... We're multiplying the identity matrix times a very general matrix. I'm just even speaking in generalities about these dimensions. Well one thing we know is that matrix multiplication is only defined is if the column, the number of columns of the first matrix is equal to the number of rows of the second matrix. This one has a rows, so this one's going to have a columns. Now how many rows is this one going to have? We already know that matrix multiplication is only defined if the number of columns on the first matrix is equal to the number of rows on the second one. We know that the product gets its number of rows from the number of rows of the first matrix being multiplied. The product has a rows then the identity matrix right over here has to have a rows. What's interesting about this? When we first got introduced to identity matrices, we were multiplying, we picked out a three by three example and we got a three by three identity matrix. What's interesting about what we've just proven to ourselves is the identity matrix for any matrix, even a non square matrix, a and b could be two different values. The identity matrix for any matrix is going to be a square matrix. It's going to have the same number of rows and the same number of columns. When we think about identity matrices, we can really just say, well is this the identity matrix that is a four by four? Is it a three by three? Is it a two by two? Or I guess one by one? The convention is, it isn't even to write identity two by two is equal to one, zero, zero, one. The convention is actually just write I2 because you know it's going to be a two by two. It's going to be a two by two matrix, it's going to be one, zero, zero, one. Identity five is going to be a five by five matrix. It's going to be one, one, two, three, four. Zero, one, two, one, three. Zero, zero, one, zero, zero. Zero ... you get the idea, zero, zero, zero, one, zero. Zero, zero, zero, zero, one. Just like that. The whole point here is just to realize that your identity matrix is always going to be a square matrix and it works even when you're multiplying non square other matrices.