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### Course: Precalculus>Unit 7

Lesson 11: Properties of matrix multiplication

# Intro to identity matrix

Just as any number remains the same when multiplied by 1, any matrix remains the same when multiplied by the identity matrix. Learn more from Sal. Created by Sal Khan.

## Want to join the conversation?

What are matrices used for?
• Also A. And matrices are used for a lot of things. Like computer programming and stuff like that. You can also use it to represent equations. For example:
3x+4y=2
6x+2y=9
Then you can represent it in matrix:
[ 3 4 [ x [2
6 2 ] * y ] = 9]
And then solving it by time each side by its inverse.
Then you will get something like:
[x [a
y] = b]
Yay! So you will know x=a and y=b

Another way you can use matrices is for formula for triangle's area. Which is pretty neat because you just put in the points of your triangle. Like (1,2), (3,0) , (4,5). Into the formula and you will get area. :VVV

And computer programming is what you might be needing it the most since all the things listed above can be done in some other ways too. In computer programming, matrices is un avoidable.

OK, I think I wrote too much unorganized facts. >.< But hey, it's really helpful. So learn it well, cause you will need it.
• Can you divide a matrix by a matrix? If so, and if it follows standard division, than Matrix I has to be equal to 1. Is this correct? at ?
• Matrix arithmetic doesn't have division, but it has inverses, which is really the same thing.
A matrix multiplied by its inverse (if it has one) gives an identity matrix.
• I took AI and got A again. It seems that with an identity matrix, reversing the order in the operation produces the same matrix. Thus, AI = IA where I is the Identity matrix of A. I just want to make sure I did not make an error somewhere. I am trying to avoid an identity crisis :-) Thanks!
• Yes, That is right, but only for square matrices. For rectangular matrices, each matrix will have two inverse matrices, which means AI will NOT equal IA. For a square matrix, AI=IA though. Hope that helped!
• At , he mentions the dot product.. But what exactly is the dot product?
• For ordered tuples of equal length(http://en.wikipedia.org/wiki/Tuple) the dot product is defined to be the product of the corresponding terms and then the sum of those products.
Ex. For (2,1) and (3,5), their dot product is: (2)(3) + (1)(5) = 6 + 5 = 11.
Ex. For (1,2,3) and (3,4,5), their dot product is: (1)(3) + (2)(4) + (3)(5) = 3 + 8 + 15 = 26

The dot product is more useful when it comes to vectors (see Sal's videos), but it can apply to anything such as these tuples (or the groups Sal makes in this videos).

Sal says "dot product" over and over because it is quicker than saying the definition I gave above.
• Do only square matrices ( same number of rows as columns) have identity matrices? He only used examples with a 3 by 3, 4 by 4, and, a 2 by 2
• That is correct. Only nxn matrices (square ones) have an Identity Matrix.
• Hi, I am studying for a Masters in Economics and in Econometrics we use some math where the lecturer mentioned 'idempotent matrices'. I know it's different to identity matrices but from what I have read about idempotent matrices, e.g. product of a matrix multiplied by itself is the matrix itself. In essence, PP or P^2 = P. However, I do not get the relation when he has used the construction of an error term in the classical linear regression model to get:

Ehat = Y - Yhat = Y - XBetahat = Y - X (X'X) ^-1 (X')Y = Y - PY

then Y - PY = (I-P) Y

with I: identity matrix
P' = X (X'X)^ -1 X' = P
The apostrophe being 'prime' or transpose

Questions:
Why (X'X)' is X'X again?
What P represents? (PY is called a projection matrix)
What is a residual maker/annihilator matrix?

I understand this is highly specialised, as in applied to a different concept however I am totally lost. I know I have to look at the ranks of a matrix before trying to understand this, I will do so. But if you could provide any insight that would be extremely helpful.

Thank you :)
• I fully understood the concept. However, what would be the use of an identity matrix? What's useful about a matrix that returns the same matrix it multiplies?
• The same use that the number `1` has in multiplication, if you stop to think about, you constantly use the fact that `1·a = a` to solve all kind of math problems, but because it's such a basic concept you don't stop to wonder at it.

In Linear Algebra the identity matrix serves the same function, and as such it's incredibly useful, from helping you solve systems of equations to finding the inverse of matrices.
• if multiplying by the identity matrix is the equivalent to multiplying a number by one then what is its use. when where and how do we use the identity matrix?
• The use of the identity matrix will become clear to you if you continue your study. It functions like any other identity element, like 1 for multiplication and 0 for addition. In that sense, multiplying a matrix by the real scalar 1 is not the same thing as multiplying by the identity matrix .
• do identity matrices only exist for square matrices?
• No, you can create an identity matrix for a 3x2 matrix. But the identity Matrices are square matrices. This is necessary to maintain the dimensions of the non-identity matrix.
• What do these identity matrices do specificaly in real life?
(1 vote)
• Identity matrices are useful in a variety of applications in mathematics, engineering, and computer science. Here are a few examples:

In linear algebra, identity matrices are used to represent linear transformations that do not change the underlying vector space. They are often used as a starting point for solving linear systems of equations, and they are essential in defining the inverse of a matrix.

In computer graphics, identity matrices are used to represent the default position and orientation of an object or camera. They can be combined with other matrices to create complex transformations such as translations, rotations, and scaling.

In cryptography, identity matrices are used in the construction of encryption algorithms. For example, the Advanced Encryption Standard (AES) uses an identity matrix as part of its key schedule.

In probability theory, identity matrices are used to represent the identity operator on a Hilbert space. This is important in quantum mechanics, where the identity operator plays a central role in defining observables and states.

Overall, identity matrices are a fundamental mathematical concept that has many practical applications in various fields.