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### Course: Algebra (all content)>Unit 20

Lesson 9: Properties of matrix multiplication

# Zero matrix & matrix multiplication

Just as any number multiplied by zero is zero, there is a zero matrix such that any matrix multiplied by it results in that zero matrix. Learn more from Sal. Created by Sal Khan.

## Want to join the conversation?

• Couldn't you just multiply the matrix by 0?
• That would make a zero matrix indeed. It is one and the same thing.
• There are multiple zero matrices that you could use. So the zero matrix is not unique, right?

E.g. in the last example any matrix of zeroes with 3 rows would work.
• Interesting ... but not quite right ... for the last example any zero matrix with 2 columns can be used. The number of rows will dictate the number of rows in the result --
e.g.s:
`[0 0] * [1 2 3] = [0 0 0] [4 5 6] [0 0] * [1 2 3] = [0 0 0] [0 0] [4 5 6] [0 0 0] [0 0] * [1 2 3] = [0 0 0] [0 0] [4 5 6] [0 0 0] [0 0] [0 0 0]`
• For a given matrix A with format `NxM`, does the zero matrix for A needs to be strictly a `MxN` matrix filled with zeros? I think the answer is yes, so it can satisfy `0 x A = A x 0 = 0`

But what about a matrix that satisfies `0 x A = 0` but do not satisfies `A x 0 = 0` ? A `PxN` matrix filled with zeros where `P != M` is one example. Is it an asymmetric zero matrix or something like that?
• Sal mentions that when he writes 0 as a zero matrix, it may have different dimensions depending on what it is multiplying, or being multiplied by. This is even "worse" than the identity matrix I, which is always square, and whose dimension is determined by the other matrix in the multiplication. If A is
``NxM``
we can multiply it (on the left) by any
``JxN``
matrix full of 0's to get a
``JxM``
result that is also full of 0's, and similarly for multiplication on the right. So when you say
``I x A = A``
or
``A x I = A``
, at least you know the dimension(s) of the identity matrix(es) if you know the dimension of A. When you say
``0 x A = 0``
, the
``0``
on the left and the
``0``
on the right may not be the same, and aren't even uniquely determined.
• btw........ is "Zero" matrix and "Null" matrix the same thing ?
• Yes - the zero matrix and the null matrix are two names for the same thing.
• is a 3 by 3 zero matrix equivalent to a 2 by 2 zero matrix?
• Well, they are both zero matrices, but they are not interchangeable. If you want to multiply a 2 x 3 matrix C by a zero matrix Z in the order ZC, you would need the
2 x 2 zero matrix. The result would be a 2 x 3 zero matrix.
The 3 x 3 zero matrix multiplied in the order ZC would be undefined.

On the other hand if you want to multiply the same matrix C by a zero matrix Z in the order CZ, you would need the 3 x 3 zero matrix. The result would be a 2 x 3 zero matrix. If you tried to multiply the order CZ using the 2 x 2 zero matrix that worked for ZC, the result would now be undefined.

By the way, you could also use a 5 x 2 zero matrix in the order ZC. The result would be a 5 x 3 zero matrix.

Hope that helps.
• Can you raise a matrix to the power of n?
• Yes, for integer n. A^n is just n copies of the matrix A multiplied together.
• Where did sal say that I(A) = A = A(I) = A? I think I might have missed that. Or is that a mistake? Is it really true that any matrix (first) times an Identity Matrix(second) equals the first matrix?
• In the first video of this section, "Identity matrix", Sal has shown that IA=A, and then at the end of the video he encouraged us viewers to test AI=A on our own.
• Can you multiply matrices with exponents, put them into equations, or use them for variables?
• you should be able too, if you watch later videos on matrices sal might talk about it IDK for sure tho
(1 vote)
• At , in the question you gave us, you multiplied the matrix (containing elements 1,2,3,4,5,6) with the zero matrix (with dimensions 3x2). Now my question is, cant the zero matrix have its dimensions as nx2? Because the matrix multiplication would still be valid, right? If so, then the matrix product would have dimensions as nx3. Therefore, unlike inverse matrix, which is unique for every matrix, there can be 'n' number of zero matrices for a given matrix. Am I correct?