## Chapter Notes Matrices and Determinants Class 12

Matrices Vectors Determinants and Linear Algebra. 24.08.2016В В· Chapter Notes - Matrices and Determinants, Class 12, Mathematics (IIT) notes for Class 12 is made by best teachers who have written some of the best books of Class 12., 24.08.2016В В· Chapter Notes - Matrices and Determinants, Class 12, Mathematics (IIT) notes for Class 12 is made by best teachers who have written some of the best books of Class 12..

### Vectors and Matrices A MIT

Matrices Vectors Determinants and Linear Algebra. 30.09.2016В В· !! "Say NO to Huge Tuition Fees" !! We are happy to announce that we have launched "Membership Plan Offer". Now students can demand any topics вЂ¦, вЂў Calculate determinants. вЂў Calculate minors and cofactors. вЂў Define and form the adjoint matrix. вЂў Define and form the inverse matrix. вЂў Define the augmented matrix. вЂў Use all the above to solve simultaneous equations. 1. INTRODUCTION In tutorial 1 on matrices you вЂ¦.

MATHEMATICS: CONCEPTS, AND FOUNDATIONS вЂ“ Vol. I - Matrices, Vectors, Determinants, and Linear Algebra - Tadao ODA В©Encyclopedia of Life Support Systems (EOLSS) basic results on determinants, systems of linear equations, eigenvalues, real symmetric вЂ¦ It begins with the fundamentals of mathematics of matrices and determinants. Matrix inversion is discussed,with an introduction of the well known reduction methods.Equation sets are viewed as vector transformations, and the conditions of their solvability are explored. Orthogonal matrices are introduced with examples showing application to many problems requiring three dimensional thinking

11вЂђ Determinants of square matrices of dimensions 4x4 and greater The methods presented for the case of 33 matrices remain valid for all greater dimensions. You must again follow the steps for cofactor expansion: Given a square matrix # and its cofactors Гњ Гќ, the determinant is obtained by following a 4.2. Determinants of linear transformations 50 4.3. Characteristic polynomials and the Cayley-Hamilton theorem 51 Chapter 5. Eigenvalues and eigenvectors 55 5.1. Eigenvalues and eigenvectors for matrices 55 5.2. Some useful facts about roots of polynomials 56 5.3. Eigenspaces and multiplicity of eigenvalues 58 5.4. Diagonalisability of square

A review of linear algebra: Applications in R Notes for a course in Psychometric Theory to accompany Psychometric Theory with Applications in R William Revelle Department of Psychology Northwestern University Evanston, Illinois USA October, 2018 1/54. Introduction Vectors Matrices Descriptive statistics Matrix Inversion Advanced topics Outline Introduction Vectors Addition, subtraction 2.14AnalysisandDesignofFeedbackControlSystems IntroductiontoMatrices DerekRowell October2002 Modernsystemdynamicsisbaseduponamatrixrepresentationofthedynamicequations

12.05.2018В В· Grooming education academy is engaged in providing qualitative commerce education to society at concessional prices. We believe in Research, reasoning and re... of second order determinants which is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows (R 1, R 2 and R 3) and three columns (C 1, C 2 and C 3) and each way вЂ¦

In matrix algebra, the inverse of a matrix is that matrix which, when multiplied by the original matrix, gives an identity matrix. The inverse of a matrix is denoted by the superscript вЂњ-1вЂќ. Hence, AAв€’1 = Aв€’1A = I A matrix must be square to have an inverse, but not all square matrices have an inverse. In some cases, the inverse does not exist. For covariance and correlation matrices 30.09.2016В В· !! "Say NO to Huge Tuition Fees" !! We are happy to announce that we have launched "Membership Plan Offer". Now students can demand any topics вЂ¦

Basics Matrixoperations Determinantofamatrix Inversematrix Eigenvaluesofamatrix Equality, Addition, Scalar multiplication Two matrices Aand B are equalif and only if they have the same size and a ij = b ij for all i,j. If Aand B are matrices of the same size then the sumof Aand B is deп¬Ѓned by C = A+B, where c ij вЂ¦ Graphics 2011/2012, 4th quarter Lecture 4: matrices, determinants. Matrices Gaussian elimination Determinants De nitions Addition and subtraction Multiplication Transpose and inverse Inverse matrices Theinverseof a matrix A is a matrix A 1 such that AA 1 = I Only square matricespossiblyhave an inverse. Note that the inverse of A 1 is A, so we have AA 1 = A 1A = I Graphics 2011/2012, 4th

BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order. 2. If B is the inverse of A, then A is also the inverse of B. (ii) Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique. (iii) Theorem 4 : If A and B are invertible matrices of same order, then Matrices and Determinants Matrix:-An arrangement numbers (real or complex )in the form of rows and columns within the brackets is called a Matrix. The numbers that form a matrix called elements of the matrix. The matrices are denoted by capital letters If a matrix has ,

Two matrices A and B are said to be equal, written A = B, if they have the same dimension and their corresponding elements are equal, i.e., a ij = b ij for all i and j. In some instances it is convenient to think of vectors as merely being special cases of matrices. Basics Matrixoperations Determinantofamatrix Inversematrix Eigenvaluesofamatrix Equality, Addition, Scalar multiplication Two matrices Aand B are equalif and only if they have the same size and a ij = b ij for all i,j. If Aand B are matrices of the same size then the sumof Aand B is deп¬Ѓned by C = A+B, where c ij вЂ¦

of second order determinants which is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows (R 1, R 2 and R 3) and three columns (C 1, C 2 and C 3) and each way вЂ¦ of second order determinants which is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows (R 1, R 2 and R 3) and three columns (C 1, C 2 and C 3) and each way вЂ¦

Matrices and Determinants Matrix:-An arrangement numbers (real or complex )in the form of rows and columns within the brackets is called a Matrix. The numbers that form a matrix called elements of the matrix. The matrices are denoted by capital letters If a matrix has , Matrices will be explored in a separate unit. However, since there are some similarities and overlap in concepts, this page provides a quick overview of what both matrices and determinants are, and how they are different. Matrices. In this section you will see the term "matrix" and the term "matrices." Matrices is the plural of matrix.

30.09.2016В В· !! "Say NO to Huge Tuition Fees" !! We are happy to announce that we have launched "Membership Plan Offer". Now students can demand any topics вЂ¦ CHAPTER 8 Matrices and Determinants Section 8.1 Matrices and Systems of Equations You should be able to use elementary row operations to produce a row-echelon form (or reduced row-echelon form) of a matrix. 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a вЂ¦

of second order determinants which is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows (R 1, R 2 and R 3) and three columns (C 1, C 2 and C 3) and each way вЂ¦ BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order. 2. If B is the inverse of A, then A is also the inverse of B. (ii) Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique. (iii) Theorem 4 : If A and B are invertible matrices of same order, then

In matrix algebra, the inverse of a matrix is that matrix which, when multiplied by the original matrix, gives an identity matrix. The inverse of a matrix is denoted by the superscript вЂњ-1вЂќ. Hence, AAв€’1 = Aв€’1A = I A matrix must be square to have an inverse, but not all square matrices have an inverse. In some cases, the inverse does not exist. For covariance and correlation matrices In matrix algebra, the inverse of a matrix is that matrix which, when multiplied by the original matrix, gives an identity matrix. The inverse of a matrix is denoted by the superscript вЂњ-1вЂќ. Hence, AAв€’1 = Aв€’1A = I A matrix must be square to have an inverse, but not all square matrices have an inverse. In some cases, the inverse does not exist. For covariance and correlation matrices

Order 3 8.4 Permutations 8.5 Determinants of Arbitrary Order 8.6 Proper-ties of Determinants 8.7 Minors and Cofactors 8.8 Evaluation of Determi-nants 8.9 Classical Adjoint 8.10 Applications to Linear Equations, CramerвЂ™s Rule 8.11 Submatrices, Minors, Principal Minors 8.12 Block Matrices and Determinants 8.13 Determinants and Volume 8.14 Determi- Matrices will be explored in a separate unit. However, since there are some similarities and overlap in concepts, this page provides a quick overview of what both matrices and determinants are, and how they are different. Matrices. In this section you will see the term "matrix" and the term "matrices." Matrices is the plural of matrix.

This method used for 3Г—3 matrices does not work for larger matrices. a a a в€’ в€’ 11 12 13a a a a 11 12 в€’ 31 a a 32 33 21 a a 22 23 a a 31 21 + + + a 32 a 22 The determinant of a 4Г—4 matrix can be calculated by finding the determinants of a group of submatrices. Given the matrix D we select any row or column. Selecting row 1 of this matrix DETERMINANTS 2.2 Properties of Determinants In this section, we will study properties determinants have and we will see how these properties can help in computing the determinant of a matrix. We will also see how these properties can give us information about matrices. 2.2.1 Determinants and Elementary Row Operations We study how performing an elementary row operation on a matrix aвЃ„ects вЂ¦

MATHEMATICS: CONCEPTS, AND FOUNDATIONS вЂ“ Vol. I - Matrices, Vectors, Determinants, and Linear Algebra - Tadao ODA В©Encyclopedia of Life Support Systems (EOLSS) basic results on determinants, systems of linear equations, eigenvalues, real symmetric вЂ¦ MATHEMATICS: CONCEPTS, AND FOUNDATIONS вЂ“ Vol. I - Matrices, Vectors, Determinants, and Linear Algebra - Tadao ODA В©Encyclopedia of Life Support Systems (EOLSS) basic results on determinants, systems of linear equations, eigenvalues, real symmetric вЂ¦

tions, inequalities, relations,) about matrices and matters relating to them. It is collected in this form for the convenience of anyone who wants a quick desktop reference . Disclaimer: The identities, approximations and relations presented here were obviously not invented but collected, borrowed and copied from a large amount of sources. These sources include similar but shorter notes CHAPTER 8 Matrices and Determinants Section 8.1 Matrices and Systems of Equations You should be able to use elementary row operations to produce a row-echelon form (or reduced row-echelon form) of a matrix. 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a вЂ¦

Matrices will be explored in a separate unit. However, since there are some similarities and overlap in concepts, this page provides a quick overview of what both matrices and determinants are, and how they are different. Matrices. In this section you will see the term "matrix" and the term "matrices." Matrices is the plural of matrix. Basics Matrixoperations Determinantofamatrix Inversematrix Eigenvaluesofamatrix Equality, Addition, Scalar multiplication Two matrices Aand B are equalif and only if they have the same size and a ij = b ij for all i,j. If Aand B are matrices of the same size then the sumof Aand B is deп¬Ѓned by C = A+B, where c ij вЂ¦

This method used for 3Г—3 matrices does not work for larger matrices. a a a в€’ в€’ 11 12 13a a a a 11 12 в€’ 31 a a 32 33 21 a a 22 23 a a 31 21 + + + a 32 a 22 The determinant of a 4Г—4 matrix can be calculated by finding the determinants of a group of submatrices. Given the matrix D we select any row or column. Selecting row 1 of this matrix 2.14AnalysisandDesignofFeedbackControlSystems IntroductiontoMatrices DerekRowell October2002 Modernsystemdynamicsisbaseduponamatrixrepresentationofthedynamicequations

Two matrices A and B are said to be equal, written A = B, if they have the same dimension and their corresponding elements are equal, i.e., a ij = b ij for all i and j. In some instances it is convenient to think of vectors as merely being special cases of matrices. Basics Matrixoperations Determinantofamatrix Inversematrix Eigenvaluesofamatrix Equality, Addition, Scalar multiplication Two matrices Aand B are equalif and only if they have the same size and a ij = b ij for all i,j. If Aand B are matrices of the same size then the sumof Aand B is deп¬Ѓned by C = A+B, where c ij вЂ¦

### Matrices National Council of Educational Research and

#1 ca foundation maths Matrices and determinants (Part. This method used for 3Г—3 matrices does not work for larger matrices. a a a в€’ в€’ 11 12 13a a a a 11 12 в€’ 31 a a 32 33 21 a a 22 23 a a 31 21 + + + a 32 a 22 The determinant of a 4Г—4 matrix can be calculated by finding the determinants of a group of submatrices. Given the matrix D we select any row or column. Selecting row 1 of this matrix, (Section 8.1: Matrices and Determinants) 8.07 3) Row Replacement (This is perhaps poorly named, since ERO types 1 and 2 may also be viewed as вЂњrow replacementsвЂќ in a literal sense.) When we solve a system using augmented matrices, вЂ¦ We can add a multiple of one row to another row. Technical Note: This combines ideas from the Row Rescaling ERO.

### Matrices and Determinants Project Bloomington IL

Chapter Notes Matrices and Determinants Class 12. This method used for 3Г—3 matrices does not work for larger matrices. a a a в€’ в€’ 11 12 13a a a a 11 12 в€’ 31 a a 32 33 21 a a 22 23 a a 31 21 + + + a 32 a 22 The determinant of a 4Г—4 matrix can be calculated by finding the determinants of a group of submatrices. Given the matrix D we select any row or column. Selecting row 1 of this matrix 30.09.2016В В· !! "Say NO to Huge Tuition Fees" !! We are happy to announce that we have launched "Membership Plan Offer". Now students can demand any topics вЂ¦.

MATHEMATICS: CONCEPTS, AND FOUNDATIONS вЂ“ Vol. I - Matrices, Vectors, Determinants, and Linear Algebra - Tadao ODA В©Encyclopedia of Life Support Systems (EOLSS) basic results on determinants, systems of linear equations, eigenvalues, real symmetric вЂ¦ (Section 8.1: Matrices and Determinants) 8.07 3) Row Replacement (This is perhaps poorly named, since ERO types 1 and 2 may also be viewed as вЂњrow replacementsвЂќ in a literal sense.) When we solve a system using augmented matrices, вЂ¦ We can add a multiple of one row to another row. Technical Note: This combines ideas from the Row Rescaling ERO

This method used for 3Г—3 matrices does not work for larger matrices. a a a в€’ в€’ 11 12 13a a a a 11 12 в€’ 31 a a 32 33 21 a a 22 23 a a 31 21 + + + a 32 a 22 The determinant of a 4Г—4 matrix can be calculated by finding the determinants of a group of submatrices. Given the matrix D we select any row or column. Selecting row 1 of this matrix 12.05.2018В В· Grooming education academy is engaged in providing qualitative commerce education to society at concessional prices. We believe in Research, reasoning and re...

(Section 8.1: Matrices and Determinants) 8.07 3) Row Replacement (This is perhaps poorly named, since ERO types 1 and 2 may also be viewed as вЂњrow replacementsвЂќ in a literal sense.) When we solve a system using augmented matrices, вЂ¦ We can add a multiple of one row to another row. Technical Note: This combines ideas from the Row Rescaling ERO A review of linear algebra: Applications in R Notes for a course in Psychometric Theory to accompany Psychometric Theory with Applications in R William Revelle Department of Psychology Northwestern University Evanston, Illinois USA October, 2018 1/54. Introduction Vectors Matrices Descriptive statistics Matrix Inversion Advanced topics Outline Introduction Vectors Addition, subtraction

Matrices and Determinants Project Due: Wednesday, Oct. 2 Worth 40 points 1 Matrices and Linear Systems An m nmatrix is a rectangular array of numbers which has mrows and ncolumns. Basics Matrixoperations Determinantofamatrix Inversematrix Eigenvaluesofamatrix Equality, Addition, Scalar multiplication Two matrices Aand B are equalif and only if they have the same size and a ij = b ij for all i,j. If Aand B are matrices of the same size then the sumof Aand B is deп¬Ѓned by C = A+B, where c ij вЂ¦

Matrices and Determinants Matrix:-An arrangement numbers (real or complex )in the form of rows and columns within the brackets is called a Matrix. The numbers that form a matrix called elements of the matrix. The matrices are denoted by capital letters If a matrix has , This method used for 3Г—3 matrices does not work for larger matrices. a a a в€’ в€’ 11 12 13a a a a 11 12 в€’ 31 a a 32 33 21 a a 22 23 a a 31 21 + + + a 32 a 22 The determinant of a 4Г—4 matrix can be calculated by finding the determinants of a group of submatrices. Given the matrix D we select any row or column. Selecting row 1 of this matrix

In matrix algebra, the inverse of a matrix is that matrix which, when multiplied by the original matrix, gives an identity matrix. The inverse of a matrix is denoted by the superscript вЂњ-1вЂќ. Hence, AAв€’1 = Aв€’1A = I A matrix must be square to have an inverse, but not all square matrices have an inverse. In some cases, the inverse does not exist. For covariance and correlation matrices (Section 8.1: Matrices and Determinants) 8.07 3) Row Replacement (This is perhaps poorly named, since ERO types 1 and 2 may also be viewed as вЂњrow replacementsвЂќ in a literal sense.) When we solve a system using augmented matrices, вЂ¦ We can add a multiple of one row to another row. Technical Note: This combines ideas from the Row Rescaling ERO

Matrices will be explored in a separate unit. However, since there are some similarities and overlap in concepts, this page provides a quick overview of what both matrices and determinants are, and how they are different. Matrices. In this section you will see the term "matrix" and the term "matrices." Matrices is the plural of matrix. In matrix algebra, the inverse of a matrix is that matrix which, when multiplied by the original matrix, gives an identity matrix. The inverse of a matrix is denoted by the superscript вЂњ-1вЂќ. Hence, AAв€’1 = Aв€’1A = I A matrix must be square to have an inverse, but not all square matrices have an inverse. In some cases, the inverse does not exist. For covariance and correlation matrices

It begins with the fundamentals of mathematics of matrices and determinants. Matrix inversion is discussed,with an introduction of the well known reduction methods.Equation sets are viewed as vector transformations, and the conditions of their solvability are explored. Orthogonal matrices are introduced with examples showing application to many problems requiring three dimensional thinking 2.14AnalysisandDesignofFeedbackControlSystems IntroductiontoMatrices DerekRowell October2002 Modernsystemdynamicsisbaseduponamatrixrepresentationofthedynamicequations

Study Notes on Matrices & Determinants for GATE 2017 Matrices and Determinates are undoubtedly one of the most scoring and high yielding topics in GATE. At least 3-4 questions are always anticipated from Matrices and Determinants making it one of the easiest and high-yielding topics in GATE. In matrix algebra, the inverse of a matrix is that matrix which, when multiplied by the original matrix, gives an identity matrix. The inverse of a matrix is denoted by the superscript вЂњ-1вЂќ. Hence, AAв€’1 = Aв€’1A = I A matrix must be square to have an inverse, but not all square matrices have an inverse. In some cases, the inverse does not exist. For covariance and correlation matrices

BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order. 2. If B is the inverse of A, then A is also the inverse of B. (ii) Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique. (iii) Theorem 4 : If A and B are invertible matrices of same order, then 11вЂђ Determinants of square matrices of dimensions 4x4 and greater The methods presented for the case of 33 matrices remain valid for all greater dimensions. You must again follow the steps for cofactor expansion: Given a square matrix # and its cofactors Гњ Гќ, the determinant is obtained by following a

Two matrices A and B are said to be equal, written A = B, if they have the same dimension and their corresponding elements are equal, i.e., a ij = b ij for all i and j. In some instances it is convenient to think of vectors as merely being special cases of matrices. Matrices and Determinants Matrix:-An arrangement numbers (real or complex )in the form of rows and columns within the brackets is called a Matrix. The numbers that form a matrix called elements of the matrix. The matrices are denoted by capital letters If a matrix has ,

30.09.2016В В· !! "Say NO to Huge Tuition Fees" !! We are happy to announce that we have launched "Membership Plan Offer". Now students can demand any topics вЂ¦ A review of linear algebra: Applications in R Notes for a course in Psychometric Theory to accompany Psychometric Theory with Applications in R William Revelle Department of Psychology Northwestern University Evanston, Illinois USA October, 2018 1/54. Introduction Vectors Matrices Descriptive statistics Matrix Inversion Advanced topics Outline Introduction Vectors Addition, subtraction

Determinants AliApplying C 1JC 1 вЂ“bC 3 and C 2JC 2 + aC 3 If each element of a row (or column) of a determinant constant of a determinant, constant multiplies of corresponding elements of other rows (or columns) are added then the determinant is unaltered. BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order. 2. If B is the inverse of A, then A is also the inverse of B. (ii) Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique. (iii) Theorem 4 : If A and B are invertible matrices of same order, then

(Section 8.1: Matrices and Determinants) 8.07 3) Row Replacement (This is perhaps poorly named, since ERO types 1 and 2 may also be viewed as вЂњrow replacementsвЂќ in a literal sense.) When we solve a system using augmented matrices, вЂ¦ We can add a multiple of one row to another row. Technical Note: This combines ideas from the Row Rescaling ERO Two matrices A and B are said to be equal, written A = B, if they have the same dimension and their corresponding elements are equal, i.e., a ij = b ij for all i and j. In some instances it is convenient to think of vectors as merely being special cases of matrices.

вЂў Calculate determinants. вЂў Calculate minors and cofactors. вЂў Define and form the adjoint matrix. вЂў Define and form the inverse matrix. вЂў Define the augmented matrix. вЂў Use all the above to solve simultaneous equations. 1. INTRODUCTION In tutorial 1 on matrices you вЂ¦ вЂў Calculate determinants. вЂў Calculate minors and cofactors. вЂў Define and form the adjoint matrix. вЂў Define and form the inverse matrix. вЂў Define the augmented matrix. вЂў Use all the above to solve simultaneous equations. 1. INTRODUCTION In tutorial 1 on matrices you вЂ¦

Study Notes on Matrices & Determinants for GATE 2017 Matrices and Determinates are undoubtedly one of the most scoring and high yielding topics in GATE. At least 3-4 questions are always anticipated from Matrices and Determinants making it one of the easiest and high-yielding topics in GATE. CHAPTER 8 Matrices and Determinants Section 8.1 Matrices and Systems of Equations You should be able to use elementary row operations to produce a row-echelon form (or reduced row-echelon form) of a matrix. 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a вЂ¦

In matrix algebra, the inverse of a matrix is that matrix which, when multiplied by the original matrix, gives an identity matrix. The inverse of a matrix is denoted by the superscript вЂњ-1вЂќ. Hence, AAв€’1 = Aв€’1A = I A matrix must be square to have an inverse, but not all square matrices have an inverse. In some cases, the inverse does not exist. For covariance and correlation matrices Two matrices A and B are said to be equal, written A = B, if they have the same dimension and their corresponding elements are equal, i.e., a ij = b ij for all i and j. In some instances it is convenient to think of vectors as merely being special cases of matrices.

Study Notes on Matrices & Determinants for GATE 2017 Matrices and Determinates are undoubtedly one of the most scoring and high yielding topics in GATE. At least 3-4 questions are always anticipated from Matrices and Determinants making it one of the easiest and high-yielding topics in GATE. Graphics 2011/2012, 4th quarter Lecture 4: matrices, determinants. Matrices Gaussian elimination Determinants De nitions Addition and subtraction Multiplication Transpose and inverse Inverse matrices Theinverseof a matrix A is a matrix A 1 such that AA 1 = I Only square matricespossiblyhave an inverse. Note that the inverse of A 1 is A, so we have AA 1 = A 1A = I Graphics 2011/2012, 4th

Week 1 вЂ“ Vectors and Matrices Richard Earl в€— Mathematical Institute, Oxford, OX1 2LB, October 2003 Abstract Algebra and geometry of vectors. The algebra of matrices. 2x2 matrices. Inverses. Determinants. Simultaneous linear equations. Standard transformations of the plane. A review of linear algebra: Applications in R Notes for a course in Psychometric Theory to accompany Psychometric Theory with Applications in R William Revelle Department of Psychology Northwestern University Evanston, Illinois USA October, 2018 1/54. Introduction Vectors Matrices Descriptive statistics Matrix Inversion Advanced topics Outline Introduction Vectors Addition, subtraction