Matrix Multiplication

1. C1 C1 C3 C3 C2 C2 R1 R1 R1 R2 R2 R2 Result in R2,C1 Result in R2,C2 (1)(-1) + (2)(3) (1)(1) + (2)(-2) (1)(2) + (2)(3) Result in R1, C3 Result in R1, C1 Result in R1, C2 Result in R2,C3 A.B = (3)(-1) + (4)(3) (3)(1) + (4)(-2) (3)(2) + (4)(3) 5 -3 8 = 9 -5 18 Matrix Multiplication To Multiply matrix A by matrix B: • Multiply each Row in matrix A by each Column in matrix B • Multiply corresponding entries and then add the resulting products

2. 2 elements or2 rows 2 elements or2 columns 3 columns , and 2 rows We had: Result: 2 rows by 3 columns A: has 2 rows, 2 columns or 2 x 2 B: has 2 rows, 3 columns or 2 x 3 By multiplying Rows from the first matrix by Columns in the second matrix: • The result will have: number of rows of A and number of columns of B. The result AB has 2 rows and 3 columns or 2 x 3. • The number of elements in per row of A, must be equal to the number of elements in per column in B, Or: Number of columns in the A = Number of Rows in B2 = 2

3. , , For the following matrices, using the multiplication of Row by Column : • Which of the following multiplication is possible • If it is possible, find the dimension of the resulting matrix a) the number of elements per row in A (3 elements, 3 columns) A.B: the number of element per column in B (3 elements, 3 rows). b) The resulting matrix will be 2 row by 1 columns or 2 x 1 a) the number of elements per row in A (3 elements, 3 columns) A.C: the number of element per column in C (3 elements, 3 rows). b) The resulting matrix will be 2 rows by 2 columns or 2 x 2 B.C: a) the number of elements per row in B (1 elements, 1 columns) B.Cis Not Possible the number of element per column in C (3 elements, 3 rows). C.A: a) the number of elements per row in C (2 elements, 3 columns) the number of element per column in A (2 elements, 2 rows). b) The resulting matrix will be 3 rows by 3 columns or 3 x 3

4. find A2, A3, A4 and A5 If: The following example will be helpful in Markov Chain section (Section 9.2).