# Lesson 6 More Arithmetic of Matrices Solidify Understanding

## Learning Focus

Examine properties of matrix addition and multiplication.

In what ways are matrix addition and multiplication similar to adding and multiplying real numbers?

## Open Up the Math: Launch, Explore, Discuss

### 1.

You will have an opportunity to examine some of the properties of matrix addition and matrix multiplication. We will restrict this work to square

The tables below define and illustrate several properties of addition and multiplication for real numbers and ask you to determine if these same properties hold for matrix addition and matrix multiplication. While the chart asks for a single example for each property, you should experiment with matrices until you are convinced that the property holds or you have found a counter-example to show that the property does not hold. Can you base your justification on more than just trying out several examples?

Associative Property of Addition | |

Examples with Real Numbers | |

Examples with |

Associative Property of Multiplication | |

Examples with Real Numbers | |

Examples with |

Commutative Property of Addition | |

Examples with Real Numbers | |

Examples with |

Commutative Property of Multiplication | |

Examples with Real Numbers | |

Examples with |

Distributive Property of Multiplication Over Addition | |

Examples with Real Numbers | |

Examples with |

In addition to the properties listed in the previous table, addition and multiplication of real numbers include properties related to the numbers

### 2.

The Additive Identity Matrix

Find values for

### 3.

The Multiplicative Identity Matrix

Find values for

Now that we have identified the additive identity and multiplicative identity for

### 4.

Finding an Additive Inverse Matrix

Find values for

### 5.

Finding a Multiplicative Inverse Matrix

Find values for

## Ready for More?

Find the inverse of this

## Takeaways

The properties of real number arithmetic can also be related to operations with matrices as follows:

Matrix addition is .

Matrix multiplication is .

The distributive property of addition over multiplication

In addition, we define the following properties for square matrices:

The additive identity matrix for

The additive inverse matrix for

The multiplicative identity matrix for

The multiplicative inverse matrix for

## Vocabulary

- associative property of addition or multiplication
- commutative property of addition or multiplication
- distributive property of multiplication over addition
- identity: additive, multiplicative
- inverse: additive, multiplicative
- matrix (properties of operations)
- properties of operations for numbers in the rational, real, or complex number systems
**Bold**terms are new in this lesson.

## Lesson Summary

In this lesson, we compared the properties of matrix addition and multiplication with the properties of addition and multiplication of real numbers, such as the associative properties, the commutative properties, and properties of identities and inverses. We found a lot of similarities, and some differences, and learned that we can find matrices that behave like

### 1.

Solve the system of equations by graphing and substitution.

#### a.

By graphing

#### b.

By substitution

### 2.

Determine if quadrilateral