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Math Advanced Math ADVANCED ENGINEERING MATHEMATICS

To check: Whether the expressions A + B , A 2 , B 2 , AB , BA , AA T , B T A , B T B , BB T , B T AB defined or not. The expressions A 2 , AB , AA T , B T A , B T B , BB T , B T AB _ are defined, whereas the expressions A + B , B 2 , BA _ are not defined. Definition used: “Addition of two matrices: The sum of two matrices A = [ a j k ] and B = [ b j k ] of the same size is written A + B and has the entries a j k + b j k obtained by adding the corresponding entries of A and B . Matrices of different sizes cannot be added.” “Multiplication of matrices: The product C = AB (in this order) of an m × n matrix A = [ a j k ] times an r × p matrix B = [ b j k ] is defined if and only if r = n and is then the m × p matrix C = [ c j k ] with entries c j k = ∑ l = 1 n a j l b l k = a j 1 b 1 k + a j 2 b 2 k + ⋯ + a j n b n k , where j = 1 , 2 , … , m and k = 1 , 2 , … , p .” Transpose of a matrix: “The transpose of an m × n matrix A = a j k is the n × m matrix A T that has the first row of A as its first column , the second row of A as its second column , and so on. Thus the transpose of A is A T = [ a k j ] .” Description: By the definition of the matrix addition, it is observed that the matrix addition is possible only if the size of the matrices is same. Given that the matrix A is a 100 × 100 matrix and the matrix B is 100 × 50 matrix. Therefore, the expression A + B is not defined. The number of rows and the number of columns of the matrix A is same and hence, multiplying A with itself is possible. But, the number of rows and the number of columns of the matrix B is not same and hence, by the definition, multiplying B with itself is not possible. The number of columns of the matrix A is 100 and the number of rows of the matrix B is 100. So, the product AB is defined. Similarly, the number of columns of the matrix B is 50 and the number of rows of the matrix A is 100. So, the product BA is not defined. Since A is square matrix, the order of both A and A T are same. Thus, the number of columns of the matrix A is same as the number of rows of A T . So, the product A A T is defined. The order of the matrix B is 100 × 50 . By the definition of the transpose, the order of the matrix B T is 50 × 100 . The number of columns of B T is 100 and the number of rows of A is also 100. Hence, the product B T A is defined. Similarly, the number of columns of B T is 100 and the number of rows of B is also 100. Hence, the product B T B is defined and so, B B T is also defined. It is known that the product AB is defined and the order is 100 × 50 . Now, observe that the number of columns of B T is 100 and the number of rows of AB is also 100. Hence, the product B T AB is defined.

To check: Whether the expressions A + B , A 2 , B 2 , AB , BA , AA T , B T A , B T B , BB T , B T AB defined or not. The expressions A 2 , AB , AA T , B T A , B T B , BB T , B T AB _ are defined, whereas the expressions A + B , B 2 , BA _ are not defined. Definition used: “Addition of two matrices: The sum of two matrices A = [ a j k ] and B = [ b j k ] of the same size is written A + B and has the entries a j k + b j k obtained by adding the corresponding entries of A and B . Matrices of different sizes cannot be added.” “Multiplication of matrices: The product C = AB (in this order) of an m × n matrix A = [ a j k ] times an r × p matrix B = [ b j k ] is defined if and only if r = n and is then the m × p matrix C = [ c j k ] with entries c j k = ∑ l = 1 n a j l b l k = a j 1 b 1 k + a j 2 b 2 k + ⋯ + a j n b n k , where j = 1 , 2 , … , m and k = 1 , 2 , … , p .” Transpose of a matrix: “The transpose of an m × n matrix A = a j k is the n × m matrix A T that has the first row of A as its first column , the second row of A as its second column , and so on. Thus the transpose of A is A T = [ a k j ] .” Description: By the definition of the matrix addition, it is observed that the matrix addition is possible only if the size of the matrices is same. Given that the matrix A is a 100 × 100 matrix and the matrix B is 100 × 50 matrix. Therefore, the expression A + B is not defined. The number of rows and the number of columns of the matrix A is same and hence, multiplying A with itself is possible. But, the number of rows and the number of columns of the matrix B is not same and hence, by the definition, multiplying B with itself is not possible. The number of columns of the matrix A is 100 and the number of rows of the matrix B is 100. So, the product AB is defined. Similarly, the number of columns of the matrix B is 50 and the number of rows of the matrix A is 100. So, the product BA is not defined. Since A is square matrix, the order of both A and A T are same. Thus, the number of columns of the matrix A is same as the number of rows of A T . So, the product A A T is defined. The order of the matrix B is 100 × 50 . By the definition of the transpose, the order of the matrix B T is 50 × 100 . The number of columns of B T is 100 and the number of rows of A is also 100. Hence, the product B T A is defined. Similarly, the number of columns of B T is 100 and the number of rows of B is also 100. Hence, the product B T B is defined and so, B B T is also defined. It is known that the product AB is defined and the order is 100 × 50 . Now, observe that the number of columns of B T is 100 and the number of rows of AB is also 100. Hence, the product B T AB is defined.

ADVANCED ENGINEERING MATHEMATICS

ADVANCED ENGINEERING MATHEMATICS

10th Edition

ISBN: 2819770198774

Author: Kreyszig

Publisher: WILEY CONS

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Chapter 7 Solutions

ADVANCED ENGINEERING MATHEMATICS

Ch. 7.1 - Equality. Give reasons why the five matrices in...Ch. 7.1 - Double subscript notation. If you write the matrix...Ch. 7.1 - Sizes. What sizes do the matrices in Examples 1,...Ch. 7.1 - Main diagonal. What is the main diagonal of A in...Ch. 7.1 - Scalar multiplication. If A in Example 2 shows the...Ch. 7.1 - If a 12 × 12 matrix A shows the distances between...Ch. 7.1 - Addition of vectors. Can you add: A row and a...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...

Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Prob. 18P Ch. 7.1 - Prob. 19P Ch. 7.1 - TEAM PROJECT. Matrices for Networks. Matrices have...Ch. 7.2 - Multiplication. Why is multiplication of matrices...Ch. 7.2 - Square matrix. What form does a 3 × 3 matrix have...Ch. 7.2 - Product of vectors. Can every 3 × 3 matrix be...Ch. 7.2 - Skew-symmetric matrix. How many different entries...Ch. 7.2 - Same questions as in Prob. 4 for symmetric...Ch. 7.2 - Triangular matrix. If U1, U2 are upper triangular...Ch. 7.2 - Idempotent matrix, defined by A2 = A. Can you find...Ch. 7.2 - Nilpotent matrix, defined by Bm = 0 for some m....Ch. 7.2 - Transposition. Can you prove (10a)–(10c) for 3 × 3...Ch. 7.2 - Transposition. (a) Illustrate (10d) by simple...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Let Showing all intermediate results, calculate...Ch. 7.2 - Prob. 21P Ch. 7.2 - Product. Write AB in Prob. 11 in terms of row and...Ch. 7.2 - Product. Calculate AB in Prob. 11 columnwise. See...Ch. 7.2 - Commutativity. Find all 2 × 2 matrices A = [ajk]...Ch. 7.2 - TEAM PROJECT. Symmetric and Skew-Symmetric...Ch. 7.2 - Production. In a production process, let N mean...Ch. 7.2 - Concert subscription. In a community of 100,000...Ch. 7.2 - Profit vector. Two factory outlets F1 and F2 in...Ch. 7.2 - TEAM PROJECT. Special Linear Transformations....Ch. 7.3 - Prob. 1P Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Prob. 5P Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Solve the linear system given explicitly or by its...Ch. 7.3 - Prob. 10P Ch. 7.3 - Prob. 11P Ch. 7.3 - Prob. 12P Ch. 7.3 - Prob. 13P Ch. 7.3 - Prob. 14P Ch. 7.3 - Prob. 15P Ch. 7.3 - Prob. 17P Ch. 7.3 - Prob. 18P Ch. 7.3 - Prob. 19P Ch. 7.3 - Prob. 20P Ch. 7.3 - Prob. 21P Ch. 7.3 - Prob. 22P Ch. 7.3 - Prob. 23P Ch. 7.3 - Prob. 24P Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Prob. 5P Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Find the rank. Find a basis for the row space....Ch. 7.4 - Prob. 8P Ch. 7.4 - Prob. 9P Ch. 7.4 - Prob. 10P Ch. 7.4 - Show the following: rank BTAT = rank AB. (Note the...Ch. 7.4 - Show the following: rank A = rank B does not imply...Ch. 7.4 - Prob. 14P Ch. 7.4 - Prob. 15P Ch. 7.4 - Prob. 16P Ch. 7.4 - Prob. 17P Ch. 7.4 - Prob. 18P Ch. 7.4 - Prob. 19P Ch. 7.4 - Prob. 20P Ch. 7.4 - Prob. 21P Ch. 7.4 - Prob. 22P Ch. 7.4 - Prob. 23P Ch. 7.4 - Prob. 24P Ch. 7.4 - Prob. 25P Ch. 7.4 - Prob. 26P Ch. 7.4 - Prob. 27P Ch. 7.4 - Prob. 28P Ch. 7.4 - Prob. 29P Ch. 7.4 - Prob. 30P Ch. 7.4 - Prob. 31P Ch. 7.4 - Prob. 32P Ch. 7.4 - Prob. 33P Ch. 7.4 - Prob. 34P Ch. 7.4 - Prob. 35P Ch. 7.7 - Prob. 1P Ch. 7.7 - Prob. 2P Ch. 7.7 - Prob. 3P Ch. 7.7 - Prob. 4P Ch. 7.7 - Prob. 5P Ch. 7.7 - Prob. 6P Ch. 7.7 - Showing the details, evaluate: Ch. 7.7 - Showing the details, evaluate: Ch. 7.7 - Showing the details, evaluate: Ch. 7.7 - Showing the details, evaluate: Ch. 7.7 - Showing the details, evaluate: Ch. 7.7 - Prob. 12P Ch. 7.7 - Prob. 13P Ch. 7.7 - Prob. 14P Ch. 7.7 - Prob. 15P Ch. 7.7 - Prob. 17P Ch. 7.7 - Prob. 18P Ch. 7.7 - Prob. 19P Ch. 7.7 - Prob. 21P Ch. 7.7 - Prob. 22P Ch. 7.7 - Prob. 23P Ch. 7.7 - Prob. 24P Ch. 7.7 - Prob. 25P Ch. 7.8 - Prob. 1P Ch. 7.8 - Prob. 2P Ch. 7.8 - Prob. 3P Ch. 7.8 - Prob. 4P Ch. 7.8 - Prob. 5P Ch. 7.8 - Prob. 6P Ch. 7.8 - Prob. 7P Ch. 7.8 - Prob. 8P Ch. 7.8 - Prob. 9P Ch. 7.8 - Prob. 10P Ch. 7.8 - Prob. 11P Ch. 7.8 - Prob. 12P Ch. 7.8 - Prob. 13P Ch. 7.8 - Prob. 14P Ch. 7.8 - Prob. 15P Ch. 7.8 - Prob. 16P Ch. 7.8 - Prob. 17P Ch. 7.8 - Prob. 18P Ch. 7.8 - Prob. 19P Ch. 7.8 - Prob. 20P Ch. 7.9 - Prob. 1P Ch. 7.9 - Prob. 2P Ch. 7.9 - Prob. 3P Ch. 7.9 - Prob. 4P Ch. 7.9 - Prob. 5P Ch. 7.9 - Prob. 6P Ch. 7.9 - Prob. 7P Ch. 7.9 - Prob. 8P Ch. 7.9 - Prob. 9P Ch. 7.9 - Prob. 10P Ch. 7.9 - Prob. 11P Ch. 7.9 - Prob. 12P Ch. 7.9 - Prob. 13P Ch. 7.9 - Prob. 14P Ch. 7.9 - Prob. 15P Ch. 7.9 - Prob. 16P Ch. 7.9 - Prob. 17P Ch. 7.9 - Prob. 18P Ch. 7.9 - Prob. 19P Ch. 7.9 - Prob. 20P Ch. 7.9 - Prob. 21P Ch. 7.9 - Prob. 22P Ch. 7.9 - Prob. 23P Ch. 7.9 - Prob. 24P Ch. 7.9 - Prob. 25P Ch. 7 - Prob. 1RQ Ch. 7 - Prob. 2RQ Ch. 7 - Prob. 3RQ Ch. 7 - Prob. 4RQ Ch. 7 - Prob. 5RQ Ch. 7 - Prob. 6RQ Ch. 7 - Prob. 7RQ Ch. 7 - Prob. 8RQ Ch. 7 - Prob. 9RQ Ch. 7 - Prob. 10RQ Ch. 7 - Prob. 11RQ Ch. 7 - Prob. 12RQ Ch. 7 - Prob. 13RQ Ch. 7 - Prob. 14RQ Ch. 7 - Prob. 15RQ Ch. 7 - Prob. 16RQ Ch. 7 - Prob. 17RQ Ch. 7 - Prob. 18RQ Ch. 7 - Prob. 19RQ Ch. 7 - Prob. 20RQ Ch. 7 - Prob. 21RQ Ch. 7 - Prob. 22RQ Ch. 7 - Prob. 23RQ Ch. 7 - Prob. 24RQ Ch. 7 - Prob. 25RQ Ch. 7 - Prob. 26RQ Ch. 7 - Prob. 27RQ Ch. 7 - Prob. 28RQ Ch. 7 - Prob. 29RQ Ch. 7 - Prob. 30RQ Ch. 7 - Prob. 31RQ Ch. 7 - Prob. 32RQ Ch. 7 - Prob. 33RQ Ch. 7 - Prob. 34RQ Ch. 7 - Prob. 35RQ

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