Matrices and Determinants

This course primarily focuses on the practical aspect of Matrices and Determinants. It begins with introducing the Matrices and Systems of Equations which includes the Definition of Matrix, Use of Matrices to solve a system of linear equations, Use of Gaussian elimination to solve a system and Use of Gauss–Jordan elimination to solve a system. The course progresses further to explain the Definition of equality of two matrices, matrix addition, and the concept of Scalar multiplication. It deals to solve of equations that involves matrices or finding matrix from a matrix equation. It also covers the topic of Matrix multiplication and the method of Applying matrix multiplication to computer graphics. It then discusses the role of Matrix Algebra with the concept of The Matrix Inverse. The course briefly includes the description of Determinants and Cramer’s Rule. The course ends with an explanation of the use of matrix inverses to solve systems of linear equations and in applied problems.

11 topics

366 lectures

Educators

Course Curriculum

Equations and Inequalities

35 videos

Linear Equations and Functions

20 videos

Linear Equations and Inequalities

28 videos

Systems of Equations and Inequalities

25 videos

Matrices and Determinants

40 videos

Quadratic Equations

10 videos

Quadratic Functions

60 videos

50 videos

Introduction to Trigonometry

44 videos

Applications of Trigonometric Functions

15 videos

Graphing Trigonometry Functions

39 videos

Matrices and Determinants Lectures

Matrices and Determinants

Add Subtract Multiply Matricesbya Scalar

In mathematics, matrix multiplication is a binary operation that takes two matrices of the same dimensions and produces a matrix whose elements are the sum, difference, product, or quotient of the elements of the two operand matrices. For example, if A is an m×n matrix and B is an n×k matrix, then their matrix product is an m×k.

Whitney Dillinger

Matrices and Determinants

Add Subtract Multiply Matricesbya Scalar

In mathematics, matrix multiplication is a binary operation that takes two matrices of the same dimensions and produces a matrix whose elements are the sum, difference, product, or quotient of the elements of the two operand matrices. For example, if A is an m×n matrix and B is an n×k matrix, then their matrix product is an m×k.

Whitney Dillinger

Matrices and Determinants

Determinants

In mathematics, the determinant is a useful value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A) or sometimes Ad. It can be viewed as the scaling factor for the transformation described by the matrix. If the matrix represents a rotation about an axis, the determinant of the matrix is the cosine of the angle of rotation of the axis.

Whitney Dillinger

Matrices and Determinants

Identityand Inverse Matrices

In linear algebra, an identity matrix, or simply identity, of size "n" is a square matrix of order "n" whose entries are all equal to 1. The term "identity matrix" is also used for the identity operator in functional analysis. The term "identity matrix" is also used to describe a special case of a matrix in statistics.

Whitney Dillinger

Matrices and Determinants

Introductionof Matrices

A matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. A matrix is usually denoted by boldface capital letters A, B, C, … or by an italicized version of the same letters in roman type. The numbers in the matrix are the matrix elements or entries. The horizontal lines dividing the matrix are called "horizontal" or "main" (or "column") "dividers" or "main diagonals" (or "column diagonals"), and the vertical lines dividing the matrix are called "vertical" or "main" (or "row") "dividers" or "main diagonals" (or "row diagonals"). The entries of the matrix are sometimes referred to as "elements" or "entries". The number of entries in each row (or column) is called the "dimension" of the matrix.

Whitney Dillinger

Matrices and Determinants

Multiply Matrices

In mathematics, matrix multiplication is a binary operation on two matrices of equal dimension (number of rows and columns) that produces a matrix of the same dimension as the two operands. The result of matrix multiplication is not commutative, that is, the order in which the two matrices are multiplied together affects the result.

Whitney Dillinger

Matrices and Determinants

Solving Equations Involving Matrices

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. A matrix can also be used to represent linear transformations, that is, generalizations of linear functions such as f(x) = 2x + 3. The identity matrix is the matrix with ones on the main diagonal and zeros elsewhere.

Whitney Dillinger

Matrices and Determinants

Solving Systems Using Inverse Matrices

In mathematics, the inverse of a matrix is another matrix that, when multiplied by the original matrix, yields the identity matrix. The inverse of a matrix "A" is denoted "A", and is defined by the condition that "A" multiplied by its inverse equals the identity matrix. For example, the inverse of a 2×2 matrix "A" is another 2×2 matrix "A", whose inverse is "A". The inverse of a matrix can be computed using the inverse matrix algorithm, or calculated numerically.

Whitney Dillinger

Matrices and Determinants

Add Subtract Multiply Matricesbya Scalar - Example 1

In mathematics, the determinant is a useful value that can be computed from the elements of a square matrix. Determinants are used to solve systems of linear equations and to express the properties of a linear transformation. The determinant of a matrix A is denoted det(A), or sometimes by |A| (analogous to the notation for absolute value).

Whitney Dillinger

Matrices and Determinants

Add Subtract Multiply Matricesbya Scalar - Example 1

In mathematics, the determinant is a useful value that can be computed from the elements of a square matrix. Determinants are used to solve systems of linear equations and to express the properties of a linear transformation. The determinant of a matrix A is denoted det(A), or sometimes by |A| (analogous to the notation for absolute value).

Whitney Dillinger

Matrices and Determinants

Determinants - Example 1

In linear algebra, a determinant is a function of numerical matrices. If M is a square matrix, then the determinant of M, denoted det(M), is a function det: M×M?R, which maps each pair of elements in the matrix to a real number. In the case of a 2×2 matrix, the determinant is also called the "adjugate" of the matrix. The determinant of a matrix is, in some sense, the characteristic polynomial of the matrix, and the determinant of the transpose is the characteristic polynomial of the transpose.

Whitney Dillinger

Matrices and Determinants

Identityand Inverse Matrices - Example 1

In linear algebra, an identity matrix of order n is a square matrix of order n that is equal to its own inverse, that is, it is its own adjoint, and therefore is also a unit matrix.

Whitney Dillinger

Matrices and Determinants

Introductionof Matrices - Example 1

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Whitney Dillinger

Matrices and Determinants

Multiply Matrices - Example 1

In mathematics, matrix multiplication is a binary operation that takes two matrices of equal dimension and combines them into a single matrix. The result of a matrix multiplication is a matrix that is the product of the two operand matrices. If the number of columns in the first matrix equals the number of rows in the second matrix, and the dimensions are otherwise compatible, the product is a rectangular matrix.

Whitney Dillinger

Matrices and Determinants

Solving Equations Involving Matrices - Example 1

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3, since it has two rows and three columns.

Whitney Dillinger

Matrices and Determinants

Solving Systems Using Inverse Matrices - Example 1

In mathematics, the inverse of a matrix A is a matrix which, when multiplied by A, yields the identity matrix. The inverse of a matrix is denoted by A^{-1} . If A is a square matrix, then its inverse is also a square matrix.

Whitney Dillinger

Matrices and Determinants

Add Subtract Multiply Matricesbya Scalar - Example 2

In mathematics, the determinant is a useful value that can be computed from the elements of a square matrix. Determinants are used to solve systems of linear equations and to express the properties of a linear transformation. The determinant of a matrix A is denoted det(A), or sometimes by |A| (analogous to the notation for absolute value).

Whitney Dillinger

Matrices and Determinants

Add Subtract Multiply Matricesbya Scalar - Example 2

In mathematics, the determinant is a useful value that can be computed from the elements of a square matrix. Determinants are used to solve systems of linear equations and to express the properties of a linear transformation. The determinant of a matrix A is denoted det(A), or sometimes by |A| (analogous to the notation for absolute value).

Whitney Dillinger

Matrices and Determinants

Determinants - Example 2

In linear algebra, a determinant is a function of numerical matrices. If M is a square matrix, then the determinant of M, denoted det(M), is a function det: M×M?R, which maps each pair of elements in the matrix to a real number. In the case of a 2×2 matrix, the determinant is also called the "adjugate" of the matrix. The determinant of a matrix is, in some sense, the characteristic polynomial of the matrix, and the determinant of the transpose is the characteristic polynomial of the transpose.

Whitney Dillinger

Matrices and Determinants

Identityand Inverse Matrices - Example 2

In linear algebra, an identity matrix of order n is a square matrix of order n that is equal to its own inverse, that is, it is its own adjoint, and therefore is also a unit matrix.

Whitney Dillinger

Matrices and Determinants

Introductionof Matrices - Example 2

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Whitney Dillinger

Matrices and Determinants

Multiply Matrices - Example 2

In mathematics, matrix multiplication is a binary operation that takes two matrices of equal dimension and combines them into a single matrix. The result of a matrix multiplication is a matrix that is the product of the two operand matrices. If the number of columns in the first matrix equals the number of rows in the second matrix, and the dimensions are otherwise compatible, the product is a rectangular matrix.

Whitney Dillinger

Matrices and Determinants

Solving Equations Involving Matrices - Example 2

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3, since it has two rows and three columns.

Whitney Dillinger

Matrices and Determinants

Solving Systems Using Inverse Matrices - Example 2

In mathematics, the inverse of a matrix A is a matrix which, when multiplied by A, yields the identity matrix. The inverse of a matrix is denoted by A^{-1} . If A is a square matrix, then its inverse is also a square matrix.

Whitney Dillinger

Matrices and Determinants

Add Subtract Multiply Matricesbya Scalar - Example 3

In mathematics, the determinant is a useful value that can be computed from the elements of a square matrix. Determinants are used to solve systems of linear equations and to express the properties of a linear transformation. The determinant of a matrix A is denoted det(A), or sometimes by |A| (analogous to the notation for absolute value).

Whitney Dillinger

Matrices and Determinants

Add Subtract Multiply Matricesbya Scalar - Example 3

In mathematics, the determinant is a useful value that can be computed from the elements of a square matrix. Determinants are used to solve systems of linear equations and to express the properties of a linear transformation. The determinant of a matrix A is denoted det(A), or sometimes by |A| (analogous to the notation for absolute value).

Whitney Dillinger

Matrices and Determinants

Determinants - Example 3

In linear algebra, a determinant is a function of numerical matrices. If M is a square matrix, then the determinant of M, denoted det(M), is a function det: M×M?R, which maps each pair of elements in the matrix to a real number. In the case of a 2×2 matrix, the determinant is also called the "adjugate" of the matrix. The determinant of a matrix is, in some sense, the characteristic polynomial of the matrix, and the determinant of the transpose is the characteristic polynomial of the transpose.

Whitney Dillinger

Matrices and Determinants

Identityand Inverse Matrices - Example 3

In linear algebra, an identity matrix of order n is a square matrix of order n that is equal to its own inverse, that is, it is its own adjoint, and therefore is also a unit matrix.

Whitney Dillinger

Matrices and Determinants

Introductionof Matrices - Example 3

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Whitney Dillinger

Matrices and Determinants

Multiply Matrices - Example 3

In mathematics, matrix multiplication is a binary operation that takes two matrices of equal dimension and combines them into a single matrix. The result of a matrix multiplication is a matrix that is the product of the two operand matrices. If the number of columns in the first matrix equals the number of rows in the second matrix, and the dimensions are otherwise compatible, the product is a rectangular matrix.

Whitney Dillinger

Matrices and Determinants

Solving Equations Involving Matrices - Example 3

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3, since it has two rows and three columns.

Whitney Dillinger

Matrices and Determinants

Solving Systems Using Inverse Matrices - Example 3

In mathematics, the inverse of a matrix A is a matrix which, when multiplied by A, yields the identity matrix. The inverse of a matrix is denoted by A^{-1} . If A is a square matrix, then its inverse is also a square matrix.

Whitney Dillinger

Matrices and Determinants

Add Subtract Multiply Matricesbya Scalar - Example 4

In mathematics, the determinant is a useful value that can be computed from the elements of a square matrix. Determinants are used to solve systems of linear equations and to express the properties of a linear transformation. The determinant of a matrix A is denoted det(A), or sometimes by |A| (analogous to the notation for absolute value).

Whitney Dillinger

Matrices and Determinants

Add Subtract Multiply Matricesbya Scalar - Example 4

In mathematics, the determinant is a useful value that can be computed from the elements of a square matrix. Determinants are used to solve systems of linear equations and to express the properties of a linear transformation. The determinant of a matrix A is denoted det(A), or sometimes by |A| (analogous to the notation for absolute value).

Whitney Dillinger

Matrices and Determinants

Determinants - Example 4

In linear algebra, a determinant is a function of numerical matrices. If M is a square matrix, then the determinant of M, denoted det(M), is a function det: M×M?R, which maps each pair of elements in the matrix to a real number. In the case of a 2×2 matrix, the determinant is also called the "adjugate" of the matrix. The determinant of a matrix is, in some sense, the characteristic polynomial of the matrix, and the determinant of the transpose is the characteristic polynomial of the transpose.

Whitney Dillinger

Matrices and Determinants

Identityand Inverse Matrices - Example 4

In linear algebra, an identity matrix of order n is a square matrix of order n that is equal to its own inverse, that is, it is its own adjoint, and therefore is also a unit matrix.

Whitney Dillinger

Matrices and Determinants

Introductionof Matrices - Example 4

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Whitney Dillinger

Matrices and Determinants

Multiply Matrices - Example 4

In mathematics, matrix multiplication is a binary operation that takes two matrices of equal dimension and combines them into a single matrix. The result of a matrix multiplication is a matrix that is the product of the two operand matrices. If the number of columns in the first matrix equals the number of rows in the second matrix, and the dimensions are otherwise compatible, the product is a rectangular matrix.

Whitney Dillinger

Matrices and Determinants

Solving Equations Involving Matrices - Example 4

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3, since it has two rows and three columns.

Whitney Dillinger

Matrices and Determinants

Solving Systems Using Inverse Matrices - Example 4

In mathematics, the inverse of a matrix A is a matrix which, when multiplied by A, yields the identity matrix. The inverse of a matrix is denoted by A^{-1} . If A is a square matrix, then its inverse is also a square matrix.

Whitney Dillinger