Solve Systems with MatricesCollege Algebra

Describing Matrices

A matrix is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Matrices are enclosed in or , and are usually named with capital letters.Examples:

𝐴 = 1 23 4 , 𝐵 =

1 2 70 −5 67 8 2

, 𝐶 =−1 30 23 1

Describing Matrices

A matrix is often referred to by its size or dimensions: 𝑚×𝑛 indicating 𝑚rows and 𝑛 columns. Matrix entries are defined first by row and then by column. For example, the entry 𝑎34 is located at row 𝑖, column 𝑗.

𝐴 =𝑎77 𝑎78 𝑎79𝑎87 𝑎88 𝑎89𝑎97 𝑎98 𝑎99

A square matrix has dimensions 𝑛×𝑛, meaning it has the same number of rows and columns.A row matrix has dimensions 1×𝑛 (one row), such as 𝐵 = 𝑏77 𝑏78 𝑏79

A column matrix has dimensions 𝑚×1 (one column), such as 𝐶 =𝑐77𝑐87

Adding and Subtracting Matrices

Given matrices 𝐴 and 𝐵 of like dimensions, addition and subtraction of 𝐴and 𝐵 will produce matrix 𝐶 or matrix 𝐷 of the same dimension.

𝐴 + 𝐵 = 𝐶 such that 𝑎34 + 𝑏34 = 𝑐34𝐴 − 𝐵 = 𝐷 such that 𝑎34 − 𝑏34 = 𝑑34

Example:

𝐴 = −2 30 1 and 𝐵 = 8 1

5 4𝐴 + 𝐵 = 6 4

5 5𝐴 − 𝐵 = −10 2

−5 −3

Multiplying a Matrix by a Scalar

Scalar multiplication involves finding the product of a constant by each entry in the matrix.

Given 𝐴 =𝑎77 𝑎78𝑎87 𝑎88 , the scalar multiple 𝑐𝐴 =

𝑐𝑎77 𝑐𝑎78𝑐𝑎87 𝑐𝑎88

Scalar multiplication is distributive. For the matrices 𝐴 and 𝐵 with scalars 𝑎and 𝑏,

𝑎 𝐴 + 𝐵 = 𝑎𝐴 + 𝑎𝐵𝑎 + 𝑏 𝐴 = 𝑎𝐴 + 𝑏𝐴

Product of Two Matrices

The product of two matrices is only possible when the inner dimensions are the same. If 𝐴 is an 𝑚×𝑟 matrix and 𝐵 is an 𝑟×𝑛 matrix, the product matrix 𝐴𝐵 is an 𝑚×𝑛 matrix.To obtain the entry in row 𝑖, column 𝑗 of 𝐴𝐵, multiply row 𝑖 in 𝐴 by column 𝑗in 𝐵 as follows:

𝑎37 𝑎38 𝑎3@ A𝑏74𝑏84𝑏B4

= 𝑎37𝑏74 + 𝑎38𝑏84 + ⋯+ 𝑎3@𝑏B4

Matrix multiplication is associative: 𝐴𝐵 𝐶 = 𝐴(𝐵𝐶)Matrix multiplication is distributive: 𝐶 𝐴 + 𝐵 = 𝐶𝐴 + 𝐶𝐵

Product of Two Matrices

Example: Multiply 𝐴 = 1 2 34 5 6 and 𝐵 =

7 108 119 12

Solution: Since 𝐴 has dimension 2×3 and 𝐵 has dimension 3×2, 𝐴𝐵 has dimension 2×2.

𝐴𝐵 = 1 7 + 2 8 + 3(9) 1 10 + 2 11 + 3(12)4 7 + 5 8 + 6(9) 4(10) + 5 11 + 6(12)

𝐴𝐵 = 50 68122 167

Systems of Equations and Matrices

A matrix can be used to represent and solve a system of equations. The coefficients of the variables and the constants become the entries in a matrix. A vertical line separates the coefficient entries from the constants; this is called an augmented matrix.Example:

G3𝑥 − 𝑦 − 𝑧 = 0𝑥 + 𝑦 = 52𝑥 − 3𝑧 = 2

is represented as3 −1 −11 1 02 0 −3

|052

Notice that all the variables line up in their own columns, and missing terms have a coefficient of 0.

Row Operations and Row-Echelon Form

In order to solve the system of equations, we convert the augmented matrix to row-echelon form in which there are ones down the main diagonal, and zeros in every position below the main diagonal.

1 𝑎 𝑏0 1 𝑐0 0 1

We use row operations to obtain a new matrix that is row-equivalent.1. Interchange rows. (Notation: 𝑅3 ↔ 𝑅4)2. Multiply a row by a constant. (Notation: 𝑐𝑅3)3. Add the product of a row multiplied by a constant to another row.

(Notation: 𝑅3 + 𝑐𝑅4)

Gaussian Elimination

The Gaussian elimination method refers to a strategy to obtain the reduced row-echelon form of a matrix.

Example: 1 23 −2|

10−2

−3𝑅7 + 𝑅8 = 𝑅81 20 −8|

10−32 Obtain a zero in row 2, column 1

−7O𝑅8 = 𝑅8

1 20 1|

104 Obtain a one in row 2, column 2

−2𝑅8 + 𝑅7 = 𝑅71 00 1|

24 Obtain a zero in row 1, column 2

Gaussian Elimination

Given a system of equations, create the augmented matrix from the coefficients and constants. Reduce the matrix to row-echelon form to obtain the solution for the system.For example:

P 𝑥 + 2𝑦 = 103𝑥 − 2𝑦 = −2 is written as 1 2

3 −2|10−2

Reduce this matrix to 1 00 1|

24

Rewrite the system as P𝑥 = 2𝑦 = 4 for the solution of 2,4

Identity Matrix and Multiplicative Inverse

The identity matrix, 𝐼@, is a square matrix containing ones down the main diagonal and zeros everywhere else.

𝐼8 =1 00 1 𝐼9 =

1 0 00 1 00 0 1

If 𝐴 is an 𝑛×𝑛 matrix and 𝐵 is an 𝑛×𝑛 matrix such that 𝐴𝐵 = 𝐵𝐴 = 𝐼@, then 𝐵 = 𝐴S7, the multiplicative inverse of matrix 𝐴.

Example: 𝐴 = 1 5−2 −9 , 𝐵 = −9 −5

2 1

𝐴𝐵 = 1 −9 + 5(2) 1 −5 + 5(1)−2 −9 − 9(2) −2 −5 − 9(1) = 1 0

0 1 = 𝐼8

Find the Inverse Using Matrix Multiplication

To find the inverse of a given matrix, multiply it by a matrix containing unknown constants and set it equal to the identity. This will result in systems of equations that can be used to find the unknowns.

Example: 𝐴 = 1 −22 −3

1 −22 −3 A 𝑎 𝑏

𝑐 𝑑 = 1 00 1 Set up 𝐴 A 𝐴S7 = 𝐼

1𝑎 − 2𝑐 1𝑏 − 2𝑑2𝑎 − 3𝑐 2𝑏 − 3𝑑 = 1 0

0 1 Find the product

P1𝑎 − 2𝑐 = 12𝑎 − 3𝑐 = 0 , P1𝑏 − 2𝑑 = 0

2𝑏 − 3𝑑 = 1 Create systems and solve for 𝑎, 𝑐 and 𝑏, 𝑑

Find the Inverse by Augmenting with the Identity

When matrix 𝐴 is transformed into 𝐼, the augmented matrix 𝐼 transforms into 𝐴S7.

Example: 𝐴 = 2 15 3

2 15 3|

1 00 1 Augment 𝐴 with the identity

1 00 1|

3 −1−5 2 Perform row operations to turn 𝐴 into the identity

𝐴S7 = 3 −1−5 2 Inverse is the right side of the augmented matrix

Inverse of a 2×2 Matrix

If 𝐴 is a 2×2 matrix such as 𝐴 = 𝑎 𝑏𝑐 𝑑 , the multiplicative inverse of 𝐴 is

given by the formula𝐴S7 =

1𝑎𝑑 − 𝑏𝑐

𝑑 −𝑏−𝑐 𝑎

If 𝑎𝑑 − 𝑏𝑐 = 0, then 𝐴 has no inverse.

Example: 𝐴 = 1 −22 −3

Solution: 𝐴S7 = 77 S9 S(S8)(8)

−3 2−2 1 = −3 2

−2 1

Solve a System of Equations Using an Inverse

Given a system of equations, write the coefficient matrix 𝐴, the variable matrix 𝑋, and the constant matrix 𝐵. Then 𝐴𝑋 = 𝐵. Multiply both sides by the inverse of 𝐴 to obtain the solution, 𝑋 = 𝐴S7𝐵.

Example: Solve the system of equations P 3𝑥 + 8𝑦 = 54𝑥 + 11𝑦 = 7

3 84 11

𝑥𝑦 = 5

7 Write the system in matrix terms

𝐴S7 = 11 −8−4 3 Use the formula for the inverse of a 2×2 matrix

𝑥𝑦 = 11 −8

−4 357 = 11 5 + −8 7

−4 5 + 3(7) = −11 The solution is −1,1

Quick Review

• What are the dimensions of a matrix?• How do you find the product of two matrices?• What is an augmented matrix?• What are the characteristics of a matrix in row-echelon form?• What are the three row operations used to obtain row-echelon form?• How do you find the inverse of a 2×2 matrix?• Can the inverse be found for all dimensions of a matrix?• How do you solve a system of equations using Gaussian elimination?• How do you solve a system using the inverse of a matrix?