Mathematics Chapter 3: Matrices

• 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.

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.