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CALCULUS – II Matrix Multiplication by Dr. Eman Saad & Dr. Shorouk Ossama
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References Robert Wrede and Murrary R. Spiegel, Theory and Problems of Advanced Calculas, 2 nd Edition, 2002.
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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.
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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
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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
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Example: Find AB if: We have 2x3 3x2 2x2
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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
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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.
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(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
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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)
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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.
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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
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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
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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.
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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).
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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
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Example:
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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
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Problem:
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Thanks
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