Please do Part A and C and please show step by step and explain Let's show how this formula works in a specific case. Suppose A is a3 x 3 matrix and B is a 3 x 2 matrix as in our previous example, then theresult of the product AB is a 3 x 2 matrix that we can call C. Now supposewe want to find the entry on the third row in the second column of C, thenwe would compute:3€3,2 = 93,kbk,2k=1=a3,101,2 +03,2b2,2 + a3,3b3,2-Sure enough, when we look at the long version we wrote earlier for theproduct AB our result matches the entry on the second row, third column.The above formula makes it possible to calculate individual matrix ele-ments, without having to compute the entire matrix. Exercise 11.2.3.(a) Let the entries of A and B be given by aij1 ≤i, j≤ 50. Let C = AB. Compute C7,11.(b) Let the entries of A and B be given by aij1 ≤i, j≤ 22. Let C = AB. Compute c5,4.== 2i+j and bij = 2−(i+j) for= 3i+j and bij==4-(i+j) for=(c) Let the entries of A and B be given by ajrit and bij = s-(i+1) for1 ≤i, j≤ N, where r and s are arbitrary real numbers. Let C = AB.Give a general formula for cij, 1 ≤ i, j≤ N. (Note the same formulaworks if r and s are taken as complex numbers.)

Answered: Exercise 11.2.3. (a) Let the entries of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Please do Part A and C and please show step by step and explain

Let's show how this formula works in a specific case. Suppose A is a
3 x 3 matrix and B is a 3 x 2 matrix as in our previous example, then the
result of the product AB is a 3 x 2 matrix that we can call C. Now suppose
we want to find the entry on the third row in the second column of C, then
we would compute:
3
€3,2 = 93,kbk,2
k=1
=a3,101,2 +03,2b2,2 + a3,3b3,2-
Sure enough, when we look at the long version we wrote earlier for the
product AB our result matches the entry on the second row, third column.
The above formula makes it possible to calculate individual matrix ele-
ments, without having to compute the entire matrix.

Exercise 11.2.3.
(a) Let the entries of A and B be given by aij
1 ≤i, j≤ 50. Let C = AB. Compute C7,11.
(b) Let the entries of A and B be given by aij
1 ≤i, j≤ 22. Let C = AB. Compute c5,4.
=
= 2i+j and bij = 2−(i+j) for
= 3i+j and bij
=
=
4-(i+j) for
=
(c) Let the entries of A and B be given by ajrit and bij = s-(i+1) for
1 ≤i, j≤ N, where r and s are arbitrary real numbers. Let C = AB.
Give a general formula for cij, 1 ≤ i, j≤ N. (Note the same formula
works if r and s are taken as complex numbers.)

Expert Solution

Step 1 Part (a)

Let A be 50 x 50 matrix and B be 50 x 50 matrix

A =[ (a_ij)] B = [(b_ij)]

a_i,j = 2^i+j b_i,j= 2^-^(i+j)

C = AB is 50 x 50 matrix

The (7,11) th entry of C or c_{7, 11} is determined by sum of products of elements of 7 th row of A with the corresponding elements of 11th column of B

c_{7, 11} = a_{7, 1}b_{1 ,11} + a_7,2 b_{2, 11}+ a_7,3 b_{3, 11} +......... + a_{7, 50}b_{50, 11}

c_{7, 11} = $\sum_{k = 1}^{50} a_{7, k} b_{k, 11}$

$= \sum_{k = 1}^{50} 2^{7 + k} 2^{- (k + 11)} = \sum_{k = 1}^{50} 2^{7} 2^{k} 2^{- k} 2^{- 11} = \sum_{k = 1}^{50} 2^{7 - 11} 2^{k - k} = \sum_{k = 1}^{50} 2^{- 4} 2^{0} = \sum_{k = 1}^{50} 2^{- 4} = 50 \times 2^{- 4} = \frac{50}{2^{4}} = \frac{50}{16} = \frac{25}{8}$