Matrix Algebra.pdf - UNIVERSITY OF TECHNOLOGY, JAMAICA...

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Compiled by: Ms. Tiffany Givans AY 2019/20 SEM21UNIVERSITY OF TECHNOLOGY, JAMAICASCHOOL OF BUILDING AND LAND MANAGEMENTCONSTRUCTION ENGINEERING DIVISIONSTRUCTURES 4|COE4001TOPIC:MATRIX STRUCTURAL ANALYSISSUBTOPIC: MATRIX ALGEBRAINTRODUCTIONIn matrix methods of structural analysis, the fundamental relationships ofequilibrium, compatibility, and member force–displacement relations areexpressed in the form of matrix equations, and the analytical procedures areformulated by applying various matrix operations.Therefore, familiarity with the basic concepts of matrix algebra is a prerequisiteto understanding matrix structural analysis.DEFINITION OF A MATRIXA matrix is defined as a rectangular array of quantities arranged in rows andcolumns. A matrix with m rows and n columns can be expressed as follows:’
Compiled by: Ms. Tiffany Givans AY 2019/20 SEM22As shown matrices are denoted either by boldface letters (A) or by italic lettersenclosed within brackets ([A]).The quantities forming a matrix is referred to as itselements. The elements ofa matrix are usually numbers, but they can be symbols, equations, or evenother matrices (called submatrices). Each element of a matrix is represented bya double-subscripted letter, with the first subscript identifying the row and thesecond subscript identifying the column in which the element is located. Thus,above A23represents the element located in the second row and third column ofmatrixA. In general,Aijrefers to an element located in theith row andjthcolumn of matrixA.The size of a matrix is measured by the number of its rows and columns and isreferred to as theorderof the matrix. Thus, matrixAabove, which has m rowsand n columns, is of orderm × n(mbyn). As an example, consider a matrixDgiven by𝐷 = ൦35378−60122327−9−1The order of this matrix is4 × 3, and its elements are symbolically denoted byDijwithi= 1 to 4 andj= 1 to 3;for example, D13= 37,D31= 12,D42= −9, etc.TYPES OF MATRIXSome of the common types of matrices in the following paragraphs.Column Matrix (Vector)If all the elements of a matrix are arranged in a single column (i.e.,n= 1), it iscalled acolumn matrix. Column matrices are usually referred to as vectorsand are sometimes denoted by italic letters enclosed within braces. An exampleof a column matrix or vector is given by
Compiled by: Ms. Tiffany Givans AY 2019/20 SEM23Row MatrixA matrix with all its elements arranged in a single row (i.e.,m= 1) is referred toas arow matrix. For example,Square MatrixIf a matrix has the same number of rows and columns (i.e.,m = n), it is called asquare matrix. An example of a4 × 4square matrixes is given byThemain diagonalof a square matrix extends from the upper left corner to thelower right corner, and it contains elements with matching subscripts—that is,A11, A22, A33, . . ., Ann. The elements forming the main diagonal are referred toas thediagonal elements, the remaining elements of a square matrix are calledtheoff-diagonal elements.

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Term
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Matrices, Diagonal matrix, Matrix Structural Analysis, Ms Tiffany Givans, matrix methods of structural analysis

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