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## Precalculus

### Course: Precalculus>Unit 7

Lesson 11: Properties of matrix multiplication

# Intro to identity matrices

Learn what an identity matrix is and about its role in matrix multiplication.

#### What you should be familiar with before taking this lesson

A matrix is a rectangular arrangement of numbers into rows and columns.
The dimensions of a matrix tell the number of rows and columns of the matrix in that order. Since matrix $A$ has $2$ rows and $3$ columns, it is called a $2×3$ matrix.
If this is new to you, we recommend that you check out our intro to matrices.
In matrix multiplication, each entry in the product matrix is the dot product of a row in the first matrix and a column in the second matrix.
If this is new to you, we recommend that you check out our matrix multiplication article.

## Definition of identity matrix

The $n×n$ identity matrix, denoted ${I}_{n}$, is a matrix with $n$ rows and $n$ columns. The entries on the diagonal from the upper left to the bottom right are all $1$'s, and all other entries are $0$.
For example:
${I}_{2}=\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\phantom{\rule{1em}{0ex}}{I}_{3}=\left[\begin{array}{rrr}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\phantom{\rule{1em}{0ex}}{I}_{4}=\left[\begin{array}{rrrr}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
The identity matrix plays a similar role in operations with matrices as the number $1$ plays in operations with real numbers. Let's take a look.

## Investigation: Multiplying by the identity matrix

Try a few multiplication problems involving the appropriate identity matrix.
1) ${I}_{2}=\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]$ and $A=\left[\begin{array}{rr}2& 3\\ 5& 1\end{array}\right]$.
${I}_{2}\cdot A=$

2) ${I}_{3}=\left[\begin{array}{rrr}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ and $A=\left[\begin{array}{rrr}1& 5& 4\\ 3& 2& 2\\ 4& 1& 3\end{array}\right]$.
$A\cdot {I}_{3}=$

### The conclusion

The product of any square matrix and the appropriate identity matrix is always the original matrix, regardless of the order in which the multiplication was performed! In other words, $A\cdot I=I\cdot A=A$.

## Connections to the real numbers

### Multiplicative Identities

The identity matrix $I$ plays a similar role to what the number $1$ plays in the real number system.
The number $1$The identity matrix $I$
The product of $1$ and any number $a$ is $a$. $\left(a\cdot 1=1\cdot a=a\right)$The product of a square matrix $A$ and the appropriate identity matrix $I$ is $A$. $\left(A\cdot I=I\cdot A=A\right)$

### Multiplicative Inverses

Two real numbers whose product is the multiplicative identity are called multiplicative inverses. For example, the numbers $\frac{1}{3}$ and $3$ are multiplicative inverses because $\frac{1}{3}\cdot 3=1$ and $3\cdot \frac{1}{3}=1$.
In fact, all nonzero real numbers have multiplicative inverses. But does this connection hold with matrix operations?
Consider matrices $A$ and $B$.
$A=\left[\begin{array}{rr}2& 3\\ 3& 4\end{array}\right]$ $\phantom{\rule{1em}{0ex}}$ $B=\left[\begin{array}{rr}-4& 3\\ 3& -2\end{array}\right]$
We can multiply to see that $AB={I}_{2}$ and $BA={I}_{2}$.
This means that $A$ and $B$ are multiplicative inverses.
However, as we will see, not all matrices have multiplicative inverses. This is one place where the properties of real numbers differ from the properties of matrices!

## Want to join the conversation?

• How do you find the multiplicative inverse of a matrix?
• Stay tuned
• I think this phrase is not precise:
The product of 'any matrix' and the appropriate identity matrix is always the original matrix, regardless of the order in which the multiplication was performed! In other words,
A*I=I*A=A

I think this only work when the matrix A is square matrix.

For example, we have a 3x2 matrix. To make the statement A*I=A to be true, the identity matrix need to be 2x2 matrix. But to make the statement I*A=A to be true, the identity matrix in this case need to be a 3x3 matrix. You can see these two matrices are not equal.
• You are correct. In your 3rd paragraph, you show a right identity matrix such that AI=A and a left identity matrix (IA=A). The right and left identity matrices are equal only if A is square matrix. Nice catch
• How do you find the identity matrix for a matrix that is not n by n?
• If you have a matrix that is 2x3, the identity matrix will be 3x3 and look like this:
1 0 0
0 1 0
0 0 1
If you have a 3x2, the identity will be 2x2 and look like this
1 0
0 1
You can try these out by your own example. The important part is that the multiplication is defined.
• What is the use of identity matrix? IA = A .... and why do we do that?
• It's useful when you're solving for messy calculations. For example, If you have A + NA + MA, you could just simply do A(I + N + M) which makes the calculation a lot easier. Also, the product of A and A^-1 is I so that's kinda important.
Hope this helps! If you have any questions or need help, please ask! :)
• is matrices just the plural version of a matrix?
• Yup. same as vertex has the plural vertices.
• How do you find the multiplicative inverse of a given matrix?

I guess you could create a system of equation with 4 equations and variables and solve (for a 2*2 matrix). But that seems too lengthy.
Is there an easier way?
• Make two columns, with your given matrix on the left and an identity matrix of the same size on the right.

Perform row operations on the matrices. The rule is, whatever operation you do to the left matrix, you must simultaneously do to the right matrix. e.g. if you multiply the top row of your matrix by 5, you must multiply the top row of the identity matrix by 5.

Do row operations until you have an identity matrix on the left. The matrix on the right will be the inverse of your original matrix.

This is still rather computationally intensive, but that's just a fact of life. Matrix inversion is a computationally difficult task, and there is not always a way to simplify something further.
• Can anyone give an example of a matrix without a multiplicative inverse?
I understand the idea but would like to see one and think about it.