951 9.7 Properties of Matrices 9.7 Properties of Matrices ■ Basic Definitions ■ Matrix Addition ■ Special Matrices ■ Matrix Subtraction ■ Scalar Multiplication ■ Matrix Multiplication ■ An Application of Matrix Algebra Basic Definitions In this section and the next, we discuss algebraic properties of matrices. It is customary to use capital letters to name matrices and subscript notation to name elements of a matrix, as in the following matrix A. A = E a11 a12 a13 g a1n a21 a22 a23 g a2n a31 a32 a33 g a3n f f f f am1 am2 am3 g amnU An n * n matrix is a square matrix of order n because the number of rows is equal to the number of columns. A matrix with just one row is a row matrix, and a matrix with just one column is a column matrix. Two matrices are equal if they have the same dimension and if corresponding elements, position by position, are equal. Using this definition, the matrices c 2 3 1 -5d and c 1 -5 2 3d are not equal (even though they contain the same elements and have the same dimension), because at least one pair of corresponding elements differ. The first row, first column element is a11 (read “a-sub-one-one”); the second row, third column element is a23; and in general, the ith row, jth column element is aij. EXAMPLE 1 Finding Values to Make Two Matrices Equal Find the values of the variables for which each statement is true, if possible. (a) c 2 p 1 qd = c x -1 y 0d (b) c x yd = £ 1 4 0§ SOLUTION (a) From the definition of equality given above, the only way that the statement can be true is if 2 = x, 1 = y, p = -1, and q = 0. (b) This statement can never be true because the two matrices have different dimensions. (One is 2 * 1 and the other is 3 * 1.) S Now Try Exercises 13 and 17. Matrix Addition Addition of matrices is defined as follows. Addition of Matrices To add two matrices of the same dimension, add corresponding elements. Only matrices of the same dimension can be added. It can be shown that matrix addition satisfies the commutative, associative, closure, identity, and inverse properties. (See Exercises 91 and 92.)
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