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Section 4.1 Matrix Addition and Scalar Multiplication Matrix, Dimension and Entries A m x n matrix, A is a rectangular array of real numbers with m rows and n columns. We refer to m and n as the dimensions of the matrix. The numbers that appear in the matrix are called its entries. We customarily use capital letters A , B , C ,….. for the names of matrices 3 7 52 A -6 12 37 is a 2 x 3 matrix because it has two rows and three columns. 3 -6 B 39 19 is a 4 x 1 matrix because it has four rows and one column. The entries of A are 3, 7, 52, -6, 12, and 37 The entries of B are 3, -6, 39, and 19. We refer to the entries by their row and column designation. So for matrix A the second row and first column designation would be listed as a21 and in this case that value would be a21 6 . Generically for any position in A we would have aij where i would designate the row and j would designate the column. Example 1 Find the dimension of the given matrix and identify the given entry 3 13 -6 22 a32 A 39 17 19 8 Two matrices are considered equal if they have the same dimensions and the corresponding entries are equal. Example 2 Solve for x, y, z, and w x-y 12. y-w x-z 0 w 0 0 6 Row Matrix, Column Matrix, and Square Matrix A matrix with a single row is called a row matrix or row vector. G 13 5 88 this matrix is a 1 x 3 A matrix with a single column is called a column matrix or column vector. 1 -23 this matrix is a 4 x 1 D 16 17 A matrix with the same number of rows as columns is called a square matrix 7 52 H this square matrix is a 2 x 2 12 37 Matrix Addition and Subtraction Two matrices can be added ( or subtracted) if and only if they have the same dimensions. To add (or subtract) two matrices of the same dimensions, we add (or subtract) the corresponding entries. More formally, if A and B are m x n matrices, then A + B and A - B are the m x n matrices whose entries are given by: A Bij Aij Bij A Bij Aij Bij Scalar Multiplication If A is an m x n matrix and c is a real number, then c A is the m x n matrix obtained by multiplying all the entries of A by c . Generally lowercase letters c, d , e are used to denote scalars. So the ij th entry of cA is given by cAij cAij Transposition If A is an m x n matrix, then its transpose is the n x m matrix, obtained by writing its rows as columns, so the ith row of the original matrix becomes the ith column of the transpose. We denote the transpose of the matrix A by A T 3 - 6 3 7 52 T 7 12 A A -6 12 37 52 37 Properties of Transposition If A and B are m x n matrices, then the following hold: A B T AT BT cAT cAT A T T A Example 3 0 -1 0.25 - 1 1 -1 0.5 , C 1 1 Given A 1 0 , B 0 3 - 1 - 1 2 - 1 - 1 14. A C 17*. 2 A C 20*. 2 AT 4 BT