CALCULUS – II Matrix Multiplication by Dr. Eman Saad & Dr. Shorouk Ossama

References  Robert Wrede and Murrary R. Spiegel, Theory and Problems of Advanced Calculas, 2 nd Edition, 2002.

Matrix Multiplication: We now need to define the concept of the product of two matrices. Not All Matrices Can Be Multiplied: they must have the right shape, or be conformable for multiplication to be defined. The product of A and B, in this order, is written as AB (no product sign is used), but it is only defined if the number of columns in A equals the number of rows in B. The product BA might not exist, and if it does, it will not in general be equal to AB.

Let us look at the case where A is a 1x3 matrix, which is a row vector, and B is a 3x1 matrix, which is a column vector, given by: The product AB is defined as the 1x1 matrix C given by: Here, the single remaining element is the sum of the products of corresponding elements from the row in A and the column in B, Thus the product of a 1x3 matrix and a 3x1 matrix is a 1x1 matrix, This is known as a row-on-column operation. 1x3 3x1 1x1

Suppose now that A is a 2x3 matrix and that B is a 3x2 matrix which are given by: The product AB is now a 2x2 matrix C given by: Note that each row in A 'operates' on each column in B giving four elements in the 2x2 matrix C. 2x33x2 2x3 3x2 2x2

Example: Find AB if: We have 2x3 3x2 2x2

Multiplication Rule: The element in the ith row and jth column of the product consist of the row-on-column product of the ith row A and jth column in B. Example: If A is a 5x4 matrix, B is a 4x5 matrix and C is a 6x4 matrix, which is following products are defined: AB, BA, AC, CB, (AB)C, (CB)A? AB is a 5x5 matrix BA is a 4x4 matrix AC is not defined CB is a 6x5 matrix AB is a 5x5 matrix; (AB) C is not defined CB is a 6x5 matrix; (CB) A is a 6x4 matrix

Example 1: Consider the matrices Since A is a 2 × 3 matrix and B is a 3 × 4 matrix, the product AB is a 2 × 4 matrix. To determine, for example, the entry in row2 and column 3 of AB, we single out row 2 from A and column 3 from B.

(2 · 4) + (6 · 3) + (0 · 5) = 26 The entry in row 1 and column 4 of AB is computed as follows: (1 · 3) + (2 · 1) + (4 · 2) = 13

Exercise: If: Find AB and BA Note: this example illustrates the point that AB be a zero matrix without either A or B or AB being Zero. A ( B + C ) = AB + AC (distributive law of addition) A (BC) = (AB) C (associative law of multiplication)

Special Matrices: 1.Transpose Matrix: If A is any m × n matrix, then the transpose of A, denoted by A, is defined to be the n × m matrix that results from interchanging the rows and columns of A ; that is, the first column of A is the first row of A, the second column of A is the second row of A, and so forth. Example: The following are some examples of matrices and their transposes.

Example: Find A T, B T, A + B T and AB where: And confirm that (AB) T = B T A T We see that: Note That: (AB) T = B T A T ( A + B) T = A T + B T 2x3 3x2 2x3 3x2

2.Symmetric Matrices: The square matrix is said to be symmetric if: -A = A T Since rows and columns are interchanged in the transpose, this is equivalent to a ij = a ji for elements if A = [a ij ]. Thus, Is a 3x3 symmetric matrix -A = - A T Is a skew- symmetric matrix

3.Row and Column Vectors: A row vector is a matrix one row, and a column vector is one column. The transpose of a row vector is a column vector and vice versa.

4.Diagonal Matrix: A square matrix all of whose elements off the leading diagonal Zero is called a diagonal matrix. 5.Identify Matrix: The diagonal matrix with all diagonal elements 1 called the identify or unit matrix. (AI =A, IA = A).

6.Power of Matrices: If A is a square matrix of order nxn, then we write AA as A 2, AA 2 as A 3 and so on. If A is diagonal, as in: A ⁿ = AA∙ ∙ ∙A ( n >0) n factors

Example:

Application For Multiplication: If Find the set of equations for x, y, z represented by Ax = d. The set of linear equations for x, y, z is: x – y + 2z = 2 3x + y – 4z = 1 -x + 2y + z = -1

Problem:

Thanks