In these lessons, we will learn about the identity matrix and inverse matrices.

**Related Pages**

Inverse Matrix

More Lessons on Matrices

More Lessons for Algebra

Math Worksheets

We also feature a matrix calculator that will help you to find the inverse of a 3×3 matrix. Use it to check your answers.

A square matrix, **I** is an **identity matrix** if the product of **I** and any square
matrix **A** is **A**.

i.e. **IA** = **AI** = **A**

For a 2 × 2 matrix, the identity matrix for multiplication is

When we multiply a matrix with the identity matrix, the original matrix is unchanged.

If the product of two square matrices, **P** and **Q**, is the identity matrix
then **Q** is an **inverse matrix** of **P** and **P** is the inverse matrix of **Q**.

i.e. **PQ** = **QP** = **I**

The inverse matrix of **A** is denoted by **A**^{-1}. (read as “A inverse”)

**AA**^{-1} = **A**^{-1}**A** = **I**

Note that the inverse of **A**^{-1} is **A**.

**Example:**

Given that **B** is the inverse of **A**, find the values of *x* and *y*.

**Solution:**

**AB** =

Since **B** is an inverse of **A**, we know that **AB** = **I**

1 – 2*y* = 1

2*y* = 0

*y* = 0

2*x* = 1

*x* =

**The Identity Matrix**

This video introduces the identity matrix and illustrates the properties of the identity matrix.

A n × n square matrix with a main diagonal of 1’s and all other elements 0’s is called the identity
matrix **I**_{n}.

If **A** is a m × n matrix, then **I**_{m}**A** = **A** and **AI**_{n} = **A**.

Is **A** is a n × n square matrix, then

**AI**_{n} = **I**_{n}**A** = **A**.

**Determining a 2×2 Inverse Matrix Using a Formula**

This video explains the formula used to determine the inverse of a 2x2 matrix, if one exists.

**Example:**

Determine the inverse of matrix **A**.

**How to find the inverse of a 2×2 matrix using the inverse formula?**

**Identity matrix and inverse matrix**

This matrix calculator will help you find the inverse of a 3×3 matrix.

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problem solver below to practice various math topics. Try the given examples, or type in your own
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