Lesson 3 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 $2 \times 2$ matrices.

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

$(a + b) + c = a + (b + c)$

Examples with Real Numbers

Examples with $2 \times 2$ Matrices

Associative Property of Multiplication

$(a b) c = a (b c)$

Examples with Real Numbers

Examples with $2 \times 2$ Matrices

Commutative Property of Addition

$a + b = b + a$

Examples with Real Numbers

Examples with $2 \times 2$ Matrices

Commutative Property of Multiplication

$a b = b a$

Examples with Real Numbers

Examples with $2 \times 2$ Matrices

Distributive Property of Multiplication Over Addition

$a (b + c) = a b + a c$

Examples with Real Numbers

Examples with $2 \times 2$ Matrices

In addition to the properties listed in the previous table, addition and multiplication of real numbers include properties related to the numbers $0$ and $1$ . For example, the number $0$ is referred to as the additive identity because $a + 0 = 0 + a = a$ , and the number $1$ is referred to as the multiplicative identity since $a \cdot 1 = 1 \cdot a = a$ . Once the additive and multiplicative identities have been identified, we can then define additive inverses $a$ and $- a$ since $a + - a = 0$ , and multiplicative inverses $a$ and $\frac{1}{a}$ since $a \cdot \frac{1}{a} = 1$ . To decide if these properties hold for matrix operations, we will need to determine if there is a matrix that plays the role of $0$ for matrix addition, and if there is a matrix that plays the role of $1$ for matrix multiplication.

2.

The Additive Identity Matrix

Find values for $a$ , $b$ , $c$ , and $d$ so that the matrix below that contains these variables plays the role of $0$ , or the additive identity matrix, for the following matrix addition. Will this same matrix work as the additive identity for all $2 \times 2$ matrices?

$[\begin{matrix} 3 & 1 \\ 4 & 2 \end{matrix}] + [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 3 & 1 \\ 4 & 2 \end{matrix}]$

3.

The Multiplicative Identity Matrix

Find values for $a$ , $b$ , $c$ , and $d$ so that the matrix below that contains these variables plays the role of $1$ , or the multiplicative identity matrix, for the following matrix multiplication. Will this same matrix work as the multiplicative identity for all $2 \times 2$ matrices?

$[\begin{matrix} 3 & 1 \\ 4 & 2 \end{matrix}] \cdot [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 3 & 1 \\ 4 & 2 \end{matrix}]$

Now that we have identified the additive identity and multiplicative identity for $2 \times 2$ matrices, we can search for the additive inverses and multiplicative inverses of matrices.

4.

Finding an Additive Inverse Matrix

Find values for $a$ , $b$ , $c$ , and $d$ so that the matrix below that contains these variables plays the role of the additive inverse of the first matrix. Will this same process work for finding the additive inverse of all $2 \times 2$ matrices?

$[\begin{matrix} 3 & 1 \\ 4 & 2 \end{matrix}] + [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

5.

Finding a Multiplicative Inverse Matrix

Find values for $a$ , $b$ , $c$ , and $d$ so that the matrix below that contains these variables plays the role of the multiplicative inverse of the first matrix. Will this same process work for finding the multiplicative inverse of all $2 \times 2$ matrices?

$[\begin{matrix} 3 & 1 \\ 4 & 2 \end{matrix}] \cdot [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Ready for More?

Find the inverse of this $3 \times 3$ matrix: $[\begin{matrix} 1 & 2 & 1 \\ 3 & 1 & 2 \\ 3 & 2 & 2 \end{matrix}]$

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 $a (b + c) = a b + a c$ .

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

The additive identity matrix for $[\begin{matrix} a_{11} & a_{12} & \dots \\ a_{21} & a_{22} & \dots \\ ⋮ & ⋮ & ⋱ \end{matrix}]$ is

The additive inverse matrix for $[\begin{matrix} a_{11} & a_{12} & \dots \\ a_{21} & a_{22} & \dots \\ ⋮ & ⋮ & ⋱ \end{matrix}]$ is

The multiplicative identity matrix for $[\begin{matrix} a_{11} & a_{12} & \dots \\ a_{21} & a_{22} & \dots \\ ⋮ & ⋮ & ⋱ \end{matrix}]$ is

The multiplicative inverse matrix for $[\begin{matrix} a_{11} & a_{12} & \dots \\ a_{21} & a_{22} & \dots \\ ⋮ & ⋮ & ⋱ \end{matrix}]$ can be found by

Vocabulary

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 $0$ and $1$ in the real number system.

Retrieval

1.

Solve the system of equations by graphing and substitution.

${\begin{cases} y = 3 x - 4 \\ y = - \frac{1}{2} x + 3 \end{cases}$

a.

By graphing

b.

By substitution

2.

Determine if quadrilateral $A B C D$ is a rhombus.