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

Lesson 9: Properties of matrix multiplication

# Using identity & zero matrices

Sal solves a problem where he has to determine whether unknown matrices are zero or identity to make an equation true. Created by Sal Khan.

## Want to join the conversation?

• So there's an answer to a question here that gives a good reason behind why identity matrices exist, so now I'm wondering about zero matrices. If you multiply a matrix by any zero matrix, you get another zero matrix. Always a bunch of zeros, though potentially a different number of zeros.

Why would you ever need a bunch of zeros?
• For the same reason, more or less. Imagine you have an equation such as (A+B)(C+D). None of these are zero matrices, but when you add A and B, you get a zero matrix. Now you know that no matter what C+D is, as long as the multiplication is defined, the answer is a zero matrix without ever having to do the math.

There might be other applications I'm not seeing, but I'm just starting to learn about matrices, so you'll have to forgive the oversight.
• Isn't what Sal did basically guess and check? Is there a faster way to determine the solution rather than exhaust all the possibilities?
• I wouldn't call it faster, but you could write equations to solve for scalar values of A, B, and C:
A+B+C=2
3A-5B+C=4
4A+B+3C=7
-2A+3B+2C=0

and then solve for A, B, and C, with answers of 1 meaning identity and 0 meaning, well, zero. To solve this, though, you'd probably need to solve another matrix. In this case, it's easier to do it Sal's way.
• is there any more logical approach to this question rather then just hit and trial ?
• How do you divide two matrices?
If it's the opposite of multiplication, I guess it should be the same as multiplying by power -1.
But, how do you find the matrix^-1?
• We cannot divide matrices, but as you suggested we can multiply by the inverse of a matrix. Much like how you cannot divide by 0, there are certain matrices we cannot find an inverse for. The actual process for finding an inverse involves augmenting the original matrix with the identity matrix and using row operations to transform the matrix on the left into the identity matrix which leaves the right matrix as the inverse. There are later lessons in Khan academy that give far more detailed explanations on this topic though.
• Is matrix multipcation commutativ or no
• In general, no, it isn't. Only with identity and zero matrices (and possibly some other coincidental special cases).
• my teacher said that identity matrix is used to replace division in matrix, because there are no dividing in matrix, is that right?
• Not quite. The identity matrix is used to prove that your inverse matrix (which is the matrix equivalent of division, also providing the matrix is invertible), will be the result when multiplied to your original matrix.

http://www.wolframalpha.com/input/?i=%7B%7B1,2,3%7D,%7B1,0,1%7D,%7B3,2,1%7D%7D*inverse(%7B%7B1,2,3%7D,%7B1,0,1%7D,%7B3,2,1%7D%7D)
• what if the leading diagonal is from left to right ? does properties of identity matrix still holds.and why the leading diagonal is always from right to left in a diagonal matrix. why it is not other way around.
• Unless certain elements are equal, no.

http://www.wolframalpha.com/input/?i=%7B%7B0,1%7D,%7B1,0%7D%7D*%7B%7Ba,b%7D,%7Bc,d%7D%7D%3D%7B%7B1,0%7D,%7B0,1%7D%7D*%7B%7Ba,b%7D,%7Bc,d%7D%7D