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

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

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

The following example will be helpful in Markov Chain section (Section 9.2). If: find A 2, A 3, A 4 and A 5