INTRODUCTORY LINEAR ALGEBRA/ MATRICES 2024

ALGEBRA/MATRICES 2024 Motivation One important application for matrices is solving systems of linear equations. Some of the following definitions may be viewed as 'designed for solving system of linear equations'. Some terminologies Definition. (Matrix) A matrix (plural: matrices) is a rectangular array of numbers. A horizontal unit is a row, and a vertical unit is a column. The element in the th row and the th column is the th entry of the matrix. An (read 'm by n') is a matrix with rows and columns, and is the size of the matrix. The rows are counted from the top, and the columns are counted from the left. If the size of a matrix is , we simply refer to this matrix as a A matrix with m rows and n columns. number, and no brackets are needed in this case. The set of all matrices with real entries is denoted by . A capital letter is usually used to denote a matrix, while small letters are used to denote its entries. For example, denotes an matrix with entries in which and . (We may omit the subscript specifying the size of matrix if its size is already mentioned, or its size is not important.) Exercise. Consider the following three matrices and . 1 Choose correct statement(s) from the following statements. 2 Choose correct statement(s) from the following statements. for each pair 3 Choose correct statement(s) from the following statements. Submit (a21 a22) (a11 a12 ) (a31 a32) Failed to parse (syntax error): {displaystyle }} In particular, if a matrix has the same number of rows and columns, then it has some nice properties. In view of the shape of such a matrix (square- like), we define such matrices as {{colored em|square matrices}}. {{colored definition| (Square matrix) A {{colored em|square matrix}} is a matrix with the same number of rows and columns. }} We will also introduce a term, namely {{colored em|main diagonal}}, which will be useful in some situations. {{colored definition| (Main diagonal) The {{colored em|main diagonal}} of an mathntimes n} matrix (which is a square matrix) is the collection of the th, th, , th entries. Then, we will define some types of matrices for which the definitions are related to the main diagonal. A diagonal matrix is the only type of matrix that is both upper triangular and lower triangular. 2 Choose all upper triangular matrices from the following matrices.