Matrices IB Mathematics SL

Matrices Describing Matrices Adding Matrices

Matrix (Plural: Matrices) A matrix is a rectangular array of numbers used to store information Terminology: Dimensions or “Order” (#rows x #columns) Entries Square matrices, Vectors Future Application: Matrices are tools that can be used to analyze multiple dimensions and multiple variables (not just 2 and 3)

Row and Column Vectors A matrix with one dimension equal to 1 is called a vector Below is a “column vector”

Adding Matrices To add matrices, order must be identical Add corresponding entries to create a new matrix of the same order

Add the Following Matrices

Matrices Multiplication of Matrices

Today Review Addition of Matrices Multiplication of Matrices Identity Matrices, Determinants [and Inverse Matrices]

To Multiply Matrices by a Scalar (a number) Multiply each entry by the same scalar

Matrix Multiplication Activity Handout 2: Flying Matrices Advice: Solve the problem without matrices, putting your answer in a matrix at the end Goal Create Matrices Multiply Matrices

To Multiply a Matrix by a Matrix The number of COLUMNS in the first matrix must match the number of ROWS in the second matrix Take a look at your Flying Matrices problem. Notice the location of the Feed and Calc. labels:

To Multiply a Matrix by a Matrix What was done with the ( ) row and the (40 50) column to get 30,000?

Multiplication of Matrices The number of Columns in the first matrix must match the number of ROWS in the second matrix Multiply entire ROWS in the first matrix by entire COLUMNS in the second matrix

Multiplying a 2x2 by a 2x2

Practice Multiplication of Matrices

Determining the Dimensions of the Product Write out the dimensions of the two matrices (1x2)(2x2) (2x3)(3x2) If the two middle numbers line up, then multiplication is possible (the columns in the first matrix match the rows in the second matrix) The two outer numbers give the dimensions of the product (1x2)(2x2)  (1x2) (2x3)(3x2)  (2x2)

Calculators Go to the “Matrix” Menu and scroll to “Edit” Select Matrix [A]: Enter the dimensions and the entries of the first matrix, Quit Select Matrix [B]: Enter the dimensions and the entries of the first matrix, Quit Calculate [A][B]

Matrix Topics NO Commutative Property of Multiplication Identity Matrix (for square matrices) Square matrix with all zero entries except for the top left to bottom right diagonal, which contains all ones Determinant We will analyze square matrices only We will look at 2x2 matrices first, and 3x3 matrices later

Properties of the Identity Let A be a square matrix of order NxN “I” is the Identity Matrix of order NxN The following statement is true

The Determinant The determinant “determines” whether or not a matrix is “non-singular” or “invertible”. If the determinant is non-zero, the matrix is non-singular If the determinant is zero, the matrix is singular, and has no inverse The Inverse, and the ability to calculate an inverse will be important when we use matrices to solve systems of equations in multiple variables (this is a common application of matrices)

Activity Matrix Multiplication Exercise : 3 and 9 The Determinant Exercise 11.2, Problem 1 v, ix, x, xi

Matrices Inverse Matrices

Today Review: Matrix Operations The Identity and the Determinant Finding the Inverse Matrix A -1 Calculator approaches

Matrix Operations Addition To add matrices, order must be identical Add corresponding entries to create a new matrix of the same order Multiplication by a Scalar Multiply each entry by the same scalar Multiplication by a Matrix The number of Columns in the first matrix must match the number of ROWS in the second matrix Multiply entire ROWS in the first matrix by entire COLUMNS in the second matrix Note: No Commutative Property

Determinant, Identity, Inverse of Square Matrices

The Inverse For matrix A, the Inverse matrix “A -1 ” is such that the product of the matrix and its inverse is the Identity matrix

Calculating the Inverse Matrix Derivation (on ) Though you will not be required to “re-derive” the inverse, you will have to calculate the inverse matrix for 2x2 matrices

The Determinant Note the position of the determinant in the denominator. If the determinant is equal to ZERO, then the Inverse is not defined (or “there is no Inverse”). If the determinant of a matrix is equal to 0, the matrix is called “singular” Otherwise, it is “non-singular”

Determining Determinants (requires Determination) Task 1: Exercise 11.2, Problem 2: ii, vi, x, xiv Problems 3 and 5 Task 2: Find the determinant of a 2x2 Identity matrix

Matrices 2x2 and 3x3 Inverse Matrices in the Calculator Solving Systems of Equations with Matrices

Calculating the Inverse Matrix Derivation (on ) Though you will not be required to “re-derive” the inverse, you will have to calculate the inverse matrix for 2x2 matrices

The Determinant Note the position of the determinant in the denominator. If the determinant is equal to ZERO, then the Inverse is not defined (“there is no Inverse”). If the determinant of a matrix is equal to 0, the matrix is called “singular” Otherwise, it is “non-singular”

Calculators Having created a square matrix with your Matrix option, you can… Find a Determinant Matrix  Math  ”det” Find the Inverse Matrix Calculate [A] -1 Activity: Check your answers for 2-ii and 2-vi from the homework

Determinants for 3x3 Matrices You will be expected to calculate the inverse of a 2x2 matrix by hand. The formula is in your formula packet. 3x3 inverses may be found with a calculator Create the 3x3 matrix in your Matrix menu, then use the “ -1 ” button to find the inverse. Note: The determinant can be obtained through the MATH menu. Remember: if the determinant is zero, the matrix is singular and has NO inverse.

Activity Exercise 11.2 Use your calculator to answer Problem 6 and 7, a and b on each.

Systems of Equations (or “Simultaneous Equations”) Goal: We will use 2x2 and 3x3 matrices to solve systems of equations in 2 or 3 variables

Systems of Equations (or “Simultaneous Equations”) Systems of equations can be written using matrices An example of a 2- variable system is on the right

Systems of Equations (or “Simultaneous Equations”) Systems of equations can be written using matrices An example of a 3- variable system is on the right

Practice Write each of the systems on the right in matrix form Then, solve the system using your calculator

Solving Systems of Equations You can use “Substitution” or “Gaussian Elimination” to solve systems of equations Matrices can be used to simplify the process 3-variable elimination is fairly complicated, but using eliminations in a system with 4, 5 or more variables is EXTREMELY tedious

Solving Systems of Equations Let’s go back to the first example of a two- variable system In the last line, I have simplified matrix form by defining the matrices A and B Note: this is not necessary, but the slide would be very busy if I didn’t do this

Solving for x and y Since the product of A and its inverse is the identity, the column vector containing x and y can be isolated The column vector containing x and y can be obtained by evaluating A -1 B

Solution

Summary Write the system in matrix form Calculate the product or A -1 and B Give your solutions

Practice Solve each of the systems on the right using matrices

Challenge: Solve this 6- variable system!

Other Concepts For a system of n variables, you need n equations The coefficient matrix [A] needs to be a square What if the coefficient matrix is singular? These problems can all be solved using Gaussian elimination…but it will be fairly time-consuming

Homework Exercise 11.3 Problem 1 (iii, vi, ix) Problem 4 (a, c) Next Class, Quiz on: Matrices Addition, Multiplication Determinants and the Inverse Solving 2- and 3-variable systems

Quiz Details Matrices Addition, Multiplication Determinants and the Inverse Singular vs. Non-singular You should be able to find the determinant and inverse of a 2x2 matrix by hand. For a 3x3 matrix, you may use a calculator Solving 2- and 3-variable systems WITH a calculator