Basic Operations MultiplicationDeterminants Cramer’s RuleIdentityInverses Solving Systems of equations APPENDIX

A rectangular array of numbers. What is a matrix? MATRICES: BASIC OPERATIONS Menu Appendix DONUTS EATEN MonTuesWedThurs Tom Jerry2092 Steve3135 This matrix has… 3 rows and 4 columns. We describe a matrix by It’s ROWs and COLLUMNs We need to know the dimensions of a matrix before we can do much with it.

What is a matrix? A rectangular array of numbers. A axb matrix has “a” rows and “b” columns. MATRICES: BASIC OPERATIONS Menu Appendix

First column First row Second row Third row Second column Third column Fourth column Row number Column number MATRICES: BASIC OPERATIONS Menu Appendix

MATRICES: BASIC OPERATIONS Menu Appendix What is entry 23? What is entry 32? What is entry 43? What is entry 34?

If A and B are two matrices of the same size (same dimensions), MATRICES: BASIC OPERATIONS Menu Appendix then the sum of the matrices is a matrix C=A+B whose entries are the sums of the corresponding entries of A and B

Addition, Subtraction Just add elements Just subtract elements Multiply each row by each column MATRICES: BASIC OPERATIONS Menu Appendix

Properties of matrix addition: 1.Matrix addition is commutative (order of addition does not matter) 2.Matrix addition is associative 3.Addition of the zero matrix MATRICES: BASIC OPERATIONS Menu Appendix

MATRICES: BASIC OPERATIONS Menu Appendix Add the following:

Multiplication by a scalar If A is a matrix and c is a scalar, then the product cA is a matrix whose entries are obtained by multiplying each of the entries of A by c MATRICES: MULTIPLICATION Menu Appendix

MATRICES: BASIC OPERATIONS Menu Appendix

Matrix multiplication is associative Distributive law Multiplication by identity matrix Multiplication by zero matrix MATRICES: MULTIPLICATION Menu Appendix

Here’s how multiplying matrices works: MATRICES: MULTIPLICATION Menu Appendix

Here’s how multiplying matrices works: MATRICES: MULTIPLICATION Menu Appendix

Here’s how multiplying matrices works: MATRICES: MULTIPLICATION Menu Appendix

THE RULES: The number of columns in the first matrix must be the same as the number of rows in the second 2 x 33 x 2 The resulting matrix has the dimensions of the outer numbers 2 x 2 MATRICES: MULTIPLICATION Menu Appendix

A) Can you multiply these matrices? B) If so, what are the dimensions of the resulting matrix? 2 x 2 3 x 2 2 x 2 2 x 32 x 3 3 x 2 3 x 2 3 x 2 2 x 3 2 x 2 4 x 2 2 X 3 2 X 2 3 X 3 MATRICES: MULTIPLICATION Menu Appendix

Weird Matrix Thing: If A and B are Matrices, A x B is not the same as B x A MATRICES: MULTIPLICATION Menu Appendix

MORE MATRIX RULES ASSOCIATIVE PROPERTYA(BC) = (AB)C If A B and C are Matrices… ASSOCIATIVE PROPERTY (scalar)c(AB) = (cA)B = A(cB) DISTRIBUTIVE PROPERTY (kinda) Left Distributive Prop.A(B+C) = AB + AC Right Distributive Prop.(A+B)C = AC + BC MATRICES: MULTIPLICATION Menu Appendix

Practice Problems MATRICES: MULTIPLICATION Menu Appendix 2 x 2 2 X 2 3 x 2 2 x 2 3 X 2

Practice Problems MATRICES: MULTIPLICATION Menu Appendix

The determinant of a square matrix is a number obtained in a specific manner from the matrix. For a 1x1 matrix: For a 2x2 matrix: What is a determinant? Product along red arrow minus product along blue arrow MATRICES: DETERMINANTS Menu Appendix

Product along red arrow minus product along blue arrow MATRICES: DETERMINANTS Menu Appendix Find the determinant

The determinant of a square matrix is a number obtained in a specific manner from the matrix. What is a determinant? MATRICES: DETERMINANTS Menu Appendix What if the matrix isn’t square?? Then there is NO DETERMINANT.

6.837 Linear Algebra Review Determinant of a Matrix Used for inversion If det(A) = 0, then A has no inverse Can be found using factorials, pivots, and cofactors! Lots of interpretations – for more info, take MATRICES: DETERMINANTS Menu Appendix

Consider the matrix Notice (1) A matrix is an array of numbers (2) A matrix is enclosed by square brackets The determinant of a matrix is a number The symbol for the determinant of a matrix is a pair of parallel lines Computation of larger matrices is more difficult MATRICES: DETERMINANTS Menu Appendix

For ONLY a 3x3 matrix write down the first two columns after the third column Sum of products along red arrow minus sum of products along blue arrow This technique works only for 3x3 matrices MATRICES: DETERMINANTS Menu Appendix

Sum of red terms = = 35 Sum of blue terms = 0 – = 0 Determinant of matrix A= det(A) = 35 – 0 = 35 MATRICES: DETERMINANTS Menu Appendix Find the determinant

MATRICES: CRAMER’s RULE Menu Appendix

MATRICES: IDENTITY Menu Appendix

Zero matrix: A matrix all of whose entries are zero Identity matrix: A square matrix which has ‘1’ s on the diagonal and zeros everywhere else. MATRICES: IDENTITY Menu Appendix

MATRICES: INVERSES Menu Appendix

MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solve this system using any method you choose: REDUCED ROW-ECHELON FORM MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solving a System of Equations with Matrices X Y Z To solve this equation, we will take out everything that we don’t change. MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solving a System of Equations with Matrices We want to perform linear operations like adding or subtracting rows, or multiplying/dividing a whole equation by a number. We will do this until our First three columns are 0’s and 1’s This is called REDUCED ROW-ECHELON FORM MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solving a System of Equations with Matrices Add row 1 and row 3 Add row 1 and row 2 Add row 2 and row 3 MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solving a System of Equations with Matrices Row 3 x3 plus row 1 Row 3 x3 MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solving a System of Equations with Matrices Divide the bottom row by 34 MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solving a System of Equations with Matrices Bottom row x4, subtract row 3 from row 2 Row 3 x4 MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solving a System of Equations with Matrices Divide row 2 by 4 MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solving a System of Equations with Matrices Bottom row x7, subtract row 3 from row 1 Row 3 x7 MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solving a System of Equations with Matrices Divide the top row by three MATRICES: SYSTEM OF EQUATIONS Menu Appendix

Solving a System of Equations with Matrices X Y Z MATRICES: SYSTEM OF EQUATIONS Menu Appendix

MATRICES: APPENDIX Menu Appendix OPENERS ASSIGNMENTS EXTRA PROBLEMS USING A CALCULATOR USING A CALCULATOR

MATRICES: OPENERS Menu Appendix ABCDEFGHIJKLMNOPABCDEFGHIJKLMNOP

MATRICES: OPENERS O Menu Appendix Solve this system of equations using any method you choose: Gaussian Elimination Reduced Row Echelon Form

MATRICES: ASSIGNMENTS Menu Appendix p. 203 p. 211 p Ch. 4 review booklet

MATRICES: EXTRA PROBLEMS Menu Appendix CHAPTER 4 DIGITAL STATIONS

MATRICES: EXTRA PROBLEMS Menu Appendix