Matrices

A matrix is a rectangular array of numbers or functions arranged in rows and columns. Matrices are always denoted by capital letters. A matrix with m rows and n columns is called a matrix of order m-by-n. A matrix with one row only is called a row matrix. A matrix with one column only is called a column matrix. A matrix whose all elements are zero, is called a zero matrix and it is denoted by O. A matrix having the number of rows equal to the number of columns is called a square matrix. A diagonal matrix is a square matrix with all the non-diagonal elements zero. A scalar matrix is a diagonal matrix with all the diagonal elements equal. An identity matrix is a diagonal matrix with all the diagonal elements equal to one. The sum of two m×n matrices A and B is denoted by A + B and A + B is computed by adding corresponding elements of A and B. If a matrix is multiplied by a scalar quantity k, each element of the matrix is multiplied by that scalar quantity For matrices A and B, the product AB is defined only if the number of columns in matrix A is the same as the number of rows in matrix B. Transpose of a Matrix A matrix obtained by interchanging all the rows and columns, is called transpose of the matrix. It is denoted by A’ or AT. For any matrices A and B: (i) (A′)’ = A (ii) (kA)’ = kA’ (iii) (A + B)’ = A’ + B’ (iv) (AB)’ = B’A’ A square matrix is called symmetric if it is equal to its transpose i.e. AT = A. A square matrix is called skew - symmetric if AT = – A. Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix. Elementary Transformations (i) The interchange of any two rows or two columns. (ii) The multiplication of the elements of any row or column by a non zero number. (iii) The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non-zero number. Invertible Matrix Square matrix “A” of order n is invertible, if there exists a square matrix “B” of the same order such that AB = BA = I. Here, A is said to be invertible and B is called inverse matrix of A and it is denoted by A–1. Inverse of a square matrix, if exists, is unique. Keywords: Types of Matrices, Null matrix, Unit matrix, Properties of Scalar Multiplication, Equality of Matrices, Addition of Matrices, Subtraction of Matrices, Multiplication of Matrices, Properties of Matrix Addition, Properties of Matrix Multiplication, Multiplication of a Matrix by a Scalar, Commutative Law, Associative Law, Properties of Transpose of the Matrices. Elementary Transformations on a Matrix, Symmetric and Skew-symmetric Matrices

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