Properties and Applications of Matrices 6.5 Learn matrix notation Learn how matrices are used in social networks Find sums, differences, and scalar multiples of matrices Find matrix products Use technology (optional)

Matrix Notation A general element of a matrix is denoted aij. The ij refers to an element in the ith row, jth column. For example, a31 would be an element in matrix A located in the third row, first column. The matrices are equal if corresponding elements are equal.

Example: Determining matrix elements Let aij denote a general element in A, and bij denote a general element in B, where (a) Identify a12, b32 and a13 . (b) Compute a31b13 + a32b23 + a33b33 . (c) Is there a value for x that will make the statement A = B true?

Example: Determining matrix elements Solution (a) Element a12 is located in the 1st row and 2nd column of A = 3. b32 = 2 (row 3, column 2) a13= 4 (row 1, column 3)

Example: Determining matrix elements Compute a31b13 + a32b23 + a33b33 . = (4)(7) + (2)(–2) + (5)(2) = 34 (c) No, since a32 = 2 ≠ 5 = b32 and a33 = 5 ≠ 2 = b32 . So A ≠ B.

Operations on Matrices Matrix Addition The sum of two m  n matrices A and B is the m  n matrix A + B, in which each element is the sum of the corresponding elements of A and B. This is written as A + B = [aij] + [bij] = [aij + bij]. If A and B have different dimensions, then A + B is undefined.

Operations on Matrices Matrix Subtraction The difference of two m  n matrices A and B is the m  n matrix A – B, in which each element is the difference of the corresponding elements of A and B. This is written as A – B = [aij] – [bij] = [aij – bij]. If A and B have different dimensions, then A – B is undefined.

Operations on Matrices Multiplication of a Matrix by a Scalar The product of a scalar (real number) k and an m  n matrix A is the m  n matrix kA, in which each element is k times the corresponding element of A. This is written as kA = k[aij] = [kaij].

Example: Performing operations on matrices If possible, perform the indicated operations using (a) A + 3B (b) A  C (c) –2C  3D

Example: Performing operations on matrices Solution (a) A + 3B

Example: Performing operations on matrices (b) A – C is undefined because the dimension of A is 2  2 and unequal to the dimension of C, which is 3  2.

Example: Performing operations on matrices (c) –2C  3D

Matrix Multiplication The product of an m  n matrix A and an n  k matrix B is the m  k matrix AB, which is computed as follows. To find the element of AB in the ith row and jth column, multiply each element in the ith row of A by the corresponding element in the jth column of B. The sum of these products will give the element in row i, column j of AB.

Matrix Multiplication

Example: Multiplying matrices If possible, compute each product using a) AB b) CA (c) DC

Example: Multiplying matrices Solution (a) AB =

Example: Multiplying matrices Solution (b) CA =

Example: Multiplying matrices Solution c) The dimension of D is 3  3 and the dimension of C is 2  3. Therefore DC is undefined. The number of columns in D (3) does not match the number of rows in C (2).

Properties of Matrices Let A, B, and C be matrices. Assume that each matrix operation is defined. 1. A + B = B + A 2. (A + B) + C = A + (B + C) 3. (AB)C = A(BC) 4. A(B + C) = AB + AC