Podcast
Questions and Answers
What is true about the inverse of a matrix?
What is true about the inverse of a matrix?
What is a necessary condition for the product BA to be equal to the product AB?
What is a necessary condition for the product BA to be equal to the product AB?
What is the inverse of a rectangular matrix?
What is the inverse of a rectangular matrix?
If B is the inverse of A, what is true about A?
If B is the inverse of A, what is true about A?
Signup and view all the answers
What is the formula for the inverse of the product of two invertible matrices?
What is the formula for the inverse of the product of two invertible matrices?
Signup and view all the answers
What can be said about the matrices X, A and B if X = AB?
What can be said about the matrices X, A and B if X = AB?
Signup and view all the answers
Why is the inverse of a matrix, if it exists, unique?
Why is the inverse of a matrix, if it exists, unique?
Signup and view all the answers
What is a prerequisite for the inverse of a matrix to exist?
What is a prerequisite for the inverse of a matrix to exist?
Signup and view all the answers
What is the correct statement about the inverse of a matrix?
What is the correct statement about the inverse of a matrix?
Signup and view all the answers
What can be said about the matrices A and B if AB = I?
What can be said about the matrices A and B if AB = I?
Signup and view all the answers
Study Notes
• The concept of matrices is essential in various branches of mathematics, and it simplifies work to a great extent compared to other straightforward methods.
• Matrices are not only used to represent coefficients in systems of linear equations but also have far-reaching utility in electronic spreadsheet programs, business, science, genetics, economics, sociology, modern psychology, and industrial management.
• A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
• A scalar matrix is a diagonal matrix where all diagonal elements are equal, and it can be denoted as B = [bij] n × n, where bij = 0 when i ≠ j, and bij = k when i = j, for some constant k.
• An identity matrix is a square matrix where all elements in the diagonal are 1, and the rest are 0, and it can be denoted as A = [aij] n × n, where aij = 1 if i = j, and aij = 0 if i ≠ j.
• A zero matrix is a matrix where all elements are 0, and it can be denoted as O.
• The negative of a matrix A is denoted by –A, and it is defined as –A = (–1) A.
• The difference of two matrices A and B of the same order is defined as A – B = A + (–1) B.
• The addition of matrices satisfies the following properties: commutative law (A + B = B + A), associative law (A + (B + C) = (A + B) + C), and additive identity (A + O = A).
• The properties of matrix addition include the commutative law, associative law, and additive identity.
• A matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.
• Elementary operations or transformations on a matrix include three operations due to rows and three due to columns, which are interchange of any two rows or columns, multiplication of a row or column by a non-zero number, and addition of a row or column to another row or column multiplied by a number.
• The inverse of a matrix A is denoted by A^–1, and it satisfies the property AA^–1 = A^–1A = I.
• If B is the inverse of A, then A is also the inverse of B, and A^–1 = B.
• The inverse of a square matrix, if it exists, is unique.
• If A and B are invertible matrices of the same order, then (AB)^–1 = B^–1A^–1.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of matrices, a powerful tool in mathematics, and its applications in solving system of linear equations. Learn about the evolution of matrices and their importance in various branches of mathematics.
