Learn on PengiSaxon Algebra 2Chapter 1: Lessons 1-10, Investigation 1

Lesson 5: Using Matrices to Organize Data and to Solve Problems

In this Grade 10 Saxon Algebra 2 lesson, students learn how to define and work with matrices, including understanding matrix dimensions, element addresses, and performing matrix addition, matrix subtraction, and scalar multiplication. The lesson covers key concepts such as additive inverse matrices, zero matrices, and solving matrix equations by applying inverse operations. Real-world data organization problems are used to reinforce how matrices can represent and calculate information more efficiently than tables.

Section 1

📘 Using Matrices to Organize Data and to Solve Problems

New Concept

A matrix is a rectangular array of numbers. A matrix with rr rows and cc columns is called an r×cr \times c matrix.

What’s next

Next, you will learn the basic operations for matrices, such as addition and subtraction, to start solving problems with them.

Section 2

Matrix Addition

Property

To add two matrices, AA and BB, of the same dimensions, add each element in the first matrix to the element that is in the same location in the second matrix.

[a11a12a21a22]+[b11b12b21b22]=[a11+b11a12+b12a21+b21a22+b22] \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}

[82544462]+[91462245]=[82+9154+4644+2262+45]=[17310066107] \begin{bmatrix} 82 & 54 \\ 44 & 62 \end{bmatrix} + \begin{bmatrix} 91 & 46 \\ 22 & 45 \end{bmatrix} = \begin{bmatrix} 82+91 & 54+46 \\ 44+22 & 62+45 \end{bmatrix} = \begin{bmatrix} 173 & 100 \\ 66 & 107 \end{bmatrix}
[1234]+[5678]=[1+52+63+74+8]=[681012] \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}
[4−159−1]+[−415−91]=[0000] \begin{bmatrix} 4 & -15 \\ 9 & -1 \end{bmatrix} + \begin{bmatrix} -4 & 15 \\ -9 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

Think of it like adding stats for two teams! To get the total, you just add the matching numbers from the same positions in each matrix. Just make sure the matrices are the same size, or it's like comparing apples to oranges—it won't work! This makes combining data sets super organized and quick.

Section 3

Matrix Subtraction

Property

To subtract two matrices of the same dimensions, A−BA - B, take the opposite, or additive inverse, of BB and add it to AA.

[a11a12a21a22]−[b11b12b21b22]=[a11+(−b11)a12+(−b12)a21+(−b21)a22+(−b22)] \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} - \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + (-b_{11}) & a_{12} + (-b_{12}) \\ a_{21} + (-b_{21}) & a_{22} + (-b_{22}) \end{bmatrix}

[510−200]−[−28−1512]=[510−200]+[2−815−12]=[72−5−12] \begin{bmatrix} 5 & 10 \\ -20 & 0 \end{bmatrix} - \begin{bmatrix} -2 & 8 \\ -15 & 12 \end{bmatrix} = \begin{bmatrix} 5 & 10 \\ -20 & 0 \end{bmatrix} + \begin{bmatrix} 2 & -8 \\ 15 & -12 \end{bmatrix} = \begin{bmatrix} 7 & 2 \\ -5 & -12 \end{bmatrix}
To solve X+[1234]=[10101010], find X=[10101010]−[1234]=[9876] \text{To solve } X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 10 & 10 \\ 10 & 10 \end{bmatrix}, \text{ find } X = \begin{bmatrix} 10 & 10 \\ 10 & 10 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 9 & 8 \\ 7 & 6 \end{bmatrix}

Matrix subtraction is a clever trick! Instead of actually subtracting, you flip the signs of every number in the second matrix to find its 'additive inverse' and then simply add them like usual. This method transforms a subtraction problem into an addition one, which makes calculations much simpler and helps you avoid those pesky sign errors.

Section 4

Scalar Multiplication

Property

To multiply matrix AA by scalar nn, multiply every element of AA by nn.

n[a11a12a21a22]=[n⋅a11n⋅a12n⋅a21n⋅a22] n \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} n \cdot a_{11} & n \cdot a_{12} \\ n \cdot a_{21} & n \cdot a_{22} \end{bmatrix}

3[1−250]=[3⋅13⋅(−2)3⋅53⋅0]=[3−6150] 3 \begin{bmatrix} 1 & -2 \\ 5 & 0 \end{bmatrix} = \begin{bmatrix} 3 \cdot 1 & 3 \cdot (-2) \\ 3 \cdot 5 & 3 \cdot 0 \end{bmatrix} = \begin{bmatrix} 3 & -6 \\ 15 & 0 \end{bmatrix}
If plan A fees are [1.251.752.503.00],doubling them is 2A=[2.503.505.006.00] \text{If plan A fees are } \begin{bmatrix} 1.25 & 1.75 \\ 2.50 & 3.00 \end{bmatrix}, \text{doubling them is } 2A = \begin{bmatrix} 2.50 & 3.50 \\ 5.00 & 6.00 \end{bmatrix}
12[10−4812]=[12⋅1012⋅(−4)12⋅812⋅12]=[5−246] \frac{1}{2} \begin{bmatrix} 10 & -4 \\ 8 & 12 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} \cdot 10 & \frac{1}{2} \cdot (-4) \\ \frac{1}{2} \cdot 8 & \frac{1}{2} \cdot 12 \end{bmatrix} = \begin{bmatrix} 5 & -2 \\ 4 & 6 \end{bmatrix}

A 'scalar' is just a math-whiz term for a single number. When you multiply a matrix by a scalar, you're essentially resizing it. Imagine using a photocopier's zoom—every single number inside the matrix gets multiplied by that one scalar value. It’s a powerful way to scale up or down an entire dataset at once.

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Chapter 1: Lessons 1-10, Investigation 1

  1. Lesson 1

    Lesson 1: Using Properties of Real Numbers

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  2. Lesson 2

    Lesson 2: Evaluating Expressions and Combining Like Terms

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  3. Lesson 3

    Lesson 3: Using Rules of Exponents

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  4. Lesson 4

    LAB 1: Graphing Calculator: Graphing a Function and Building a Table

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  5. Lesson 5

    Lesson 4: Identifying Functions and Using Function Notation

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  6. Lesson 6

    LAB 2: Graphing Calculator: Storing and Recalling Data in a Matrix

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  7. Lesson 7Current

    Lesson 5: Using Matrices to Organize Data and to Solve Problems

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  8. Lesson 8

    Lesson 6: Finding Percent of Change

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  9. Lesson 9

    Lesson 7: Solving Linear Equations (Exploration: Solving Equations Using Algebra Tiles)

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  10. Lesson 10

    Lesson 8: Finding Direct Variation

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  11. Lesson 11

    Lesson 9: Multiplying Matrices

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  12. Lesson 12

    Lesson 10: Solving and Graphing Inequalities

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  13. Lesson 13

    Investigation 1: Logic and Truth Tables

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Section 1

📘 Using Matrices to Organize Data and to Solve Problems

New Concept

A matrix is a rectangular array of numbers. A matrix with rr rows and cc columns is called an r×cr \times c matrix.

What’s next

Next, you will learn the basic operations for matrices, such as addition and subtraction, to start solving problems with them.

Section 2

Matrix Addition

Property

To add two matrices, AA and BB, of the same dimensions, add each element in the first matrix to the element that is in the same location in the second matrix.

[a11a12a21a22]+[b11b12b21b22]=[a11+b11a12+b12a21+b21a22+b22] \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}

[82544462]+[91462245]=[82+9154+4644+2262+45]=[17310066107] \begin{bmatrix} 82 & 54 \\ 44 & 62 \end{bmatrix} + \begin{bmatrix} 91 & 46 \\ 22 & 45 \end{bmatrix} = \begin{bmatrix} 82+91 & 54+46 \\ 44+22 & 62+45 \end{bmatrix} = \begin{bmatrix} 173 & 100 \\ 66 & 107 \end{bmatrix}
[1234]+[5678]=[1+52+63+74+8]=[681012] \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}
[4−159−1]+[−415−91]=[0000] \begin{bmatrix} 4 & -15 \\ 9 & -1 \end{bmatrix} + \begin{bmatrix} -4 & 15 \\ -9 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

Think of it like adding stats for two teams! To get the total, you just add the matching numbers from the same positions in each matrix. Just make sure the matrices are the same size, or it's like comparing apples to oranges—it won't work! This makes combining data sets super organized and quick.

Section 3

Matrix Subtraction

Property

To subtract two matrices of the same dimensions, A−BA - B, take the opposite, or additive inverse, of BB and add it to AA.

[a11a12a21a22]−[b11b12b21b22]=[a11+(−b11)a12+(−b12)a21+(−b21)a22+(−b22)] \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} - \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + (-b_{11}) & a_{12} + (-b_{12}) \\ a_{21} + (-b_{21}) & a_{22} + (-b_{22}) \end{bmatrix}

[510−200]−[−28−1512]=[510−200]+[2−815−12]=[72−5−12] \begin{bmatrix} 5 & 10 \\ -20 & 0 \end{bmatrix} - \begin{bmatrix} -2 & 8 \\ -15 & 12 \end{bmatrix} = \begin{bmatrix} 5 & 10 \\ -20 & 0 \end{bmatrix} + \begin{bmatrix} 2 & -8 \\ 15 & -12 \end{bmatrix} = \begin{bmatrix} 7 & 2 \\ -5 & -12 \end{bmatrix}
To solve X+[1234]=[10101010], find X=[10101010]−[1234]=[9876] \text{To solve } X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 10 & 10 \\ 10 & 10 \end{bmatrix}, \text{ find } X = \begin{bmatrix} 10 & 10 \\ 10 & 10 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 9 & 8 \\ 7 & 6 \end{bmatrix}

Matrix subtraction is a clever trick! Instead of actually subtracting, you flip the signs of every number in the second matrix to find its 'additive inverse' and then simply add them like usual. This method transforms a subtraction problem into an addition one, which makes calculations much simpler and helps you avoid those pesky sign errors.

Section 4

Scalar Multiplication

Property

To multiply matrix AA by scalar nn, multiply every element of AA by nn.

n[a11a12a21a22]=[n⋅a11n⋅a12n⋅a21n⋅a22] n \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} n \cdot a_{11} & n \cdot a_{12} \\ n \cdot a_{21} & n \cdot a_{22} \end{bmatrix}

3[1−250]=[3⋅13⋅(−2)3⋅53⋅0]=[3−6150] 3 \begin{bmatrix} 1 & -2 \\ 5 & 0 \end{bmatrix} = \begin{bmatrix} 3 \cdot 1 & 3 \cdot (-2) \\ 3 \cdot 5 & 3 \cdot 0 \end{bmatrix} = \begin{bmatrix} 3 & -6 \\ 15 & 0 \end{bmatrix}
If plan A fees are [1.251.752.503.00],doubling them is 2A=[2.503.505.006.00] \text{If plan A fees are } \begin{bmatrix} 1.25 & 1.75 \\ 2.50 & 3.00 \end{bmatrix}, \text{doubling them is } 2A = \begin{bmatrix} 2.50 & 3.50 \\ 5.00 & 6.00 \end{bmatrix}
12[10−4812]=[12⋅1012⋅(−4)12⋅812⋅12]=[5−246] \frac{1}{2} \begin{bmatrix} 10 & -4 \\ 8 & 12 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} \cdot 10 & \frac{1}{2} \cdot (-4) \\ \frac{1}{2} \cdot 8 & \frac{1}{2} \cdot 12 \end{bmatrix} = \begin{bmatrix} 5 & -2 \\ 4 & 6 \end{bmatrix}

A 'scalar' is just a math-whiz term for a single number. When you multiply a matrix by a scalar, you're essentially resizing it. Imagine using a photocopier's zoom—every single number inside the matrix gets multiplied by that one scalar value. It’s a powerful way to scale up or down an entire dataset at once.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 1: Lessons 1-10, Investigation 1

  1. Lesson 1

    Lesson 1: Using Properties of Real Numbers

    LearnPractice
  2. Lesson 2

    Lesson 2: Evaluating Expressions and Combining Like Terms

    LearnPractice
  3. Lesson 3

    Lesson 3: Using Rules of Exponents

    LearnPractice
  4. Lesson 4

    LAB 1: Graphing Calculator: Graphing a Function and Building a Table

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  5. Lesson 5

    Lesson 4: Identifying Functions and Using Function Notation

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  6. Lesson 6

    LAB 2: Graphing Calculator: Storing and Recalling Data in a Matrix

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  7. Lesson 7Current

    Lesson 5: Using Matrices to Organize Data and to Solve Problems

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  8. Lesson 8

    Lesson 6: Finding Percent of Change

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  9. Lesson 9

    Lesson 7: Solving Linear Equations (Exploration: Solving Equations Using Algebra Tiles)

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  10. Lesson 10

    Lesson 8: Finding Direct Variation

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  11. Lesson 11

    Lesson 9: Multiplying Matrices

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  12. Lesson 12

    Lesson 10: Solving and Graphing Inequalities

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  13. Lesson 13

    Investigation 1: Logic and Truth Tables

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