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CE 311 K - Introduction to Computer Methods Daene C. McKinneyMatrices CE 311 K - Introduction to Computer Methods Daene C. McKinney
Introduction Matrices Matrix Arithmetic Diagonal MatricesAddition Multiplication Diagonal Matrices Identity matrix Matrix Inverse
Matrix Matrix - a rectangular array of numbers arranged into m rows and n columns: Rows, i = 1, …, m Columns, j = 1, …, n
Matrices Variety of engineering problems lead to the need to solve systems of linear equations matrix column vectors
Row and Column Matrices (vectors)Row matrix (or row vector) is a matrix with one row Column vector is a matrix with one column
Square Matrix When the row and column dimensions of a matrix are equal (m = n) then the matrix is called square
Matrix Transpose The transpose of the (m x n) matrix A is the (n x m) matrix formed by interchanging the rows and columns such that row i becomes column i of the transposed matrix
Matrix Equality Two (m x n) matrices A and B are equal if and only if each of their elements are equal. That is A = B if and only if aij = bij for i = 1,...,m; j = 1,...,n
Vector Addition The sum of two (m x 1) column vectors a and b is
Matrix MultiplicationThe product of two matrices A and B is defined only if the number of columns of A is equal to the number of rows of B. If A is (m x p) and B is (p x n), the product is an (m x n) matrix C
Example - Matrix Multiplication
Diagonal Matrices Diagonal Matrix Identity MatrixThe identity matrix has the property that if A is a square matrix, then
Matrix Inverse If A is a square matrix and there is a matrix X with the property that X is defined to be the inverse of A and is denoted A-1 Example 2x2 matrix inverse
Summary Matrices Matrix Arithmetic Diagonal Matrices Matrix InverseAddition Multiplication Diagonal Matrices Identity matrix Matrix Inverse
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Matrices Addition & Subtraction Scalar Multiplication & Multiplication Determinants Inverses Solving Systems – 2x2 & 3x3 Cramer’s Rule.
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