Help Me Solve Matrix: Show B†A† = C†

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In summary, the conversation is about the relationship between matrices A and A†, where A is an m×n matrix with (i, j)-entry aij and A† is the n×m matrix with (i, j)-entry aji. The first part of the question asks to show that if the product C = AB is defined, then so is B†A†, and the second part asks to prove that B†A† = C†. The conversation ends with a request for elaboration on the answer provided by Elabed Haidar.
  • #1
theacerf1
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Please HELP! Matrix!

Please help me with the below question! I have no idea how to solve this, if someone could please help me with a solution and explain what they did and how they did it, it would be such a BIG help! Thanks! :)


If A is an m×n matrix with (i, j)-entry aij , let A† be the n×m matrix with (i, j)-entry
aji. Show that

(i) if the product C = AB is defined, then so is B†A†,
(ii) B†A† = C†.
 
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  • #2


well can you define B ??
 
  • #3


Thats all the question says, so confused :confused:
 
  • #4


since C =AB is defined then the matrix AB is defined thus B is defined to be a n x c matrix where c is any number
since A† be the n×m matrix so B† is defined as a c x n so B† A† is defined as a c x m matrix therefore B† A† is defined
 
  • #5


okay second part comes from the first part if C is AB then it is a m x c matrix use A† is an n x m matrix with (i, j)-entry aij mulitply with B† by saying that B is a c x n matrix with (i, j)-entry bij and mulitply them you will get your answer but one note i didnt give you details so you have to define each matrix
 
  • #6


can anyone elaborate on this? i still don't get Elabed Haidar's answer. Sorry
 

1. What is a matrix and how is it used in solving equations?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is used in solving equations by representing a system of equations in a compact and organized manner.

2. How do you solve for a matrix using matrix multiplication?

To solve for a matrix using matrix multiplication, you need to multiply the elements in each row of the first matrix by the corresponding elements in each column of the second matrix, and then add the products. The resulting matrix will be the solution to the equation.

3. Can you provide an example of solving for a matrix using matrix multiplication?

Sure, let's say we have the equation A = [1 2; 3 4] and B = [5 6; 7 8]. To solve for the matrix C in the equation A†B† = C†, we first find the transpose of A and B, which is A† = [1 3; 2 4] and B† = [5 7; 6 8]. Then, we multiply A† and B† to get C† = [19 25; 43 57].

4. What is the significance of the superscript "†" in the equation A†B† = C†?

The superscript "†" represents the transpose of a matrix. It indicates that the rows and columns of the matrix are switched, which is necessary for solving this type of equation.

5. Are there any other methods for solving matrix equations besides matrix multiplication?

Yes, there are other methods such as Gaussian elimination, Cramer's rule, and inverse matrices. However, matrix multiplication is often the most efficient and straightforward method for solving matrix equations.

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