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 C1 C1 C2 C2 C3 C3 R1 R1 R1 R2 R2 R2 Result in R2,C2 Result in R2,C1 Result in R1, C2 Result in R1, C1 Result in R1, C3 Result in R2,C3 A.B = (1)(-1) + (2)(3) (1)(1) + (2)(-2) (1)(2) + (2)(3) = 5 -3 8 9 -5 18 (3)(-1) + (4)(3) (3)(1) + (4)(-2) (3)(2) + (4)(3)

Number of columns in the A = Number of Rows in B 2 = 2 We had: 2 elements or 2 rows 2 elements or 2 columns 3 columns Result: 2 rows by 3 columns , and 2 rows 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 B 2 = 2

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.B: a) the number of elements per row in A (3 elements, 3 columns) 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.C: a) the number of elements per row in A (3 elements, 3 columns) 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.C is 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

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