Chapter 8: Matrices and Determinants

Chapter 8: Matrices and Determinants

Tuesday June 1: 8-1: Matrix Solutions to Linear Systems. Gauss-Jordan Elimination.

After today’s lesson you should be able to do the following:

1. Write the augmented matrix for a linear system
2. Perform matrix row operations
3. Use matrices and Guass-Jordan elimination to solve systems

Objective 1: Writing an augmented matrix given a system of equations

A matrix gives us a shortened way of writing a system of equations. The first step in solving a system of linear equations using matrices is to write the augmented matrix. An augmented matrix has a vertical bar separating the columns of the matrix into tow groups. The coefficients of each variable are placed to the left of the vertical line and the constants are placed to the right. If any variable is missing, its coefficient is 0:

In 1-2, Write the augment matrix that represents the given system of equations.

1. 2.

Objective 2: Perform matrix row operations

Examples of Row Operations

(a)Interchange the first and second Rows.

Original MatrixNew Row-Equivalent Matrix

(b)Multiply the first row by

Original MatrixNew Row-Equivalent Matrix

(c)Add −2 times the first row to the third row.

Original MatrixNew Row-Equivalent Matrix

Objective 3: Use Gauss-Jordan Elimination to solve system of equations.

With Guass-Jordan elimination, elementary row operations are applied to a matrix to obtain a row-equivalent matrix that is in a form called reduced row-echelon form. For a system of three equations a matrix in reduced row-echelon form is:

Notice that reduced row-echelon form contains a diagonal of ones with zeros in all other positions of the matrix. When you convert a matrix that is in reduced row-echelon form to a system of equations you have the solutions:

means

In 3, Use Guass-Jordan elimination to solve the system:

Wed June 2: 8-2 Inconsistent and Dependent Systems and Their Applications

After today’s lesson you should be able to do the following:

1. Apply Gaussian elimination to systems without unique solutions.
2. Apply Gaussian elimination to systems with more variables than equations.
3. Solve problems involving systems without unique solutions.

Objective 1: Apply Gaussian elimination to systems without unique solutions.

In 1, Use Gaussian elimination to solve the system:

In 2, Use Gaussian elimination to solve the system:

Objective 2: Apply Gaussian elimination to systems with more variables than equations.

When we have more variables than equations the process is similar to example #2 above when we get a true statement of 0 = 0. You need to define your solution in terms of z.

In 3, Use Gaussian elimination to solve the system:

Objective 3: Solve problems involving systems without unique solutions.

In a network, which can be a computer circuit or the intersections of streets, it is assumed that the total flow into a junction is equal to the total flow out of the junction. For example:

In 4, The flow of traffic (in vehicles per hour) through a network of streets is shown in the figure:

Thur June 3 Quiz on Gaussian Elimination to find the solution to a system of equations

Friday June 4: 8-3 Matrix Operations and Their Applications

After today’s lesson you should be able to do the following:

1. Use matrix notation
2. Understand what is meant by equal matrices
3. Add and subtract matrices
4. Perform scalar multiplication
5. Multiply matrices

Objective 1: Use matrix notation

We have seen that an array of numbers arranged in rows and columns and placed in brackets is called a matrix. We can represent a matrix in two different ways.

• A capital letter, such as A, B, or C, can denote a matrix.
• A lower case letter enclosed in brackets: can also denote a matrix.

A matrix of order m×n has m rows and n columns. If m = n the matrix is called a square matix.

Practice: (a) State the order of the matrix and (b) identify and .

Objective 2: Understand what is meant by equal matrices.

Two matrices A and B are equal if and only if they have the same order m×n and each entry in the matrix are the same.

Practice: Find the value of x, y, z given A = B: and

Objective 3: Add and subtract matrices.

Matrices of the same order can be added or subtracted by adding or subtracting the elements in corresponding positions. Matrices that are not the same order can not be added or subtracted.

Practice: Given and

Find (a) A + B(b) A – B (c) B – A

Objective 4: Perform scalar multiplication.

To multiply a matrix A by a scalar c we multiply each entry in A by c.

Practice: If and , find the following matrices:

(a) −6B(b) 3A + 2B

Practice: Solve for X in the matrix equation 3X + A = B, where and .

Objective 5: Multiply matrices.

In order to multiply matrices you must think of it as row-by-column multiplication:

Practice: Find AB given and

In order to multiply matrices AB the number of columns of A must equal the number of rows of B. The product of AB will have an order equal to the number of columns of A × number of rows of B.

Matrix AMatrix B

m × n n × p

Practice: Where possible, find each product

(a) (b)

Mon June 7: 8-4 Multiplicative Inverses of Matrices and Matrix Equations

After today’s lesson you should be able to do the following:

1. Find the multiplicative inverse of a square matrix
2. Use inverse to solve matrix equations

Objective 1: Find the multiplicative inverse of a square matrix

Remember that the multiplicative inverse of a number n has the following property: and .

We define the multiplicative inverseof a square matrix in a similar manner. However, the matrix that we use for “1” has 1s diagonal from upper let to lower right and 0s elsewhere. We call a square matrix with 1s on the diagonal and zeros elsewhere the identity matrix:

Definition of the Multiplicative Inverse of a Square Matrix

Let A be an n×n matrix. If there exists an n×n matrix A−1 (read “A inverse”) such that AA−1 = I and A−1A = I.

then A−1 is the multiplicative inverse of A.

To show that a matrix B is the multiplicative inverse of A you must show both AB = I and BA = I.

Practice: Show that B is the multiplicative inverse of A, where and .

To find the Multiplicative Inverse of a 2×2 Matrix

Practice: Find the inverse of the matrix

The above method only works for 2×2 matrices. For other square matrices you must use the following procedure.

Practice: Find the multiplicative inverse of :

Practice: Solve the system using A−1, the inverse of the coefficient matrix:

Tuesday June 8: 8-5 Determinants and Cramer’s Rule

After today’s lesson you should be able to do the following:

1. Evaluate a second-order determinant
2. Solve a system of linear equations in two variables using Cramer’s rule
3. Evaluate a third-order determinant
4. Solve a system of linear equations in three variables using Cramer’s rule

Objective 1: Evaluate a determinant of a 2×2 matrix

Practice: Evaluate the determinant of each of the following matrices:

(a) (b)

Objective 2: Solve a system of linear equations in two variables using Cramer’s rule.

Practice: Use Cramer’s rule to solve the system:

Objective 3: Evaluate a determinant of a 3×3 Matrix.

To find the determinant of a 3×3 Matrix use the following pattern:

Practice: Evaluate the determinant of: .

Objective 4: Solve a system of linear equations in three variables using Cramer’s Rule.

Step 1: Find the determinant of the coefficient matrix, called D.

Step 2: Find the determinant with the 1st column of the coefficient matrix replaced with the solutions to each linear equation, called Dx.

Step 3: Find the determinant with the 2nd column of the coefficient matrix replaced with the solutions of each linear equation, called Dy.

Step 4: Find the determinant with the 3rd column of the coefficient matrix replaced with the solutions of each linear equation, called Dz.

Step 5: Solve for x, y, and z by using the following: .

Practice: Use Cramer’s rule to solve the system:

1

Math Analysis Notes Prepared by Mr. Hopp