SOLUTION: Hi all I asked a similiar question recently and got a reaaly good response, but this imiliar problem is still giving me trouble. I need to solve the following Matrices and comment

Question 200660: Hi all I asked a similiar question recently and got a reaaly good response, but this imiliar problem is still giving me trouble.
I need to solve the following Matrices and comment on the outcomes.For the following:
A =
(2 3 0)
1 4 -3
B =
(1 -1 4)
2 3 5
C =
(2 -1)
3 5
Find and state why or why not the following is possible.
i) A+B
ii) C^-1
iii) AB
1v) B^C
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Table of Contents:
Part i)
Part ii)
Part iii)
Part iv)

Given Matrices:

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Part i)

Start with the addition of the two given matrices

Add up the matrices by adding the corresponding entries

Add

===========================================

Answer:

So

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Part ii)

To find the inverse of the matrix , we can follow these steps:

Step 1) Find the determinant

The determinant of is . So this means that

note: since the determinant is NOT equal to zero, the matrix inverse exists.

--------------------------------
Step 2) Swap the values

Now switch the highlighted values to get

--------------------------------

Step 3) Change the sign

Now change the sign of the highlighted values to get

--------------------------------

Step 4) Multiply by the inverse of the determinant

Multiply by to get

Plug in to get

--------------------------------

Step 5) Perform scalar multiplication and simplify.

Multiply by EVERY element to get

Multiply to get

=================================================================

Answer:
So the inverse of is

This means that if then

note: to verify your work, multiply the matrices and and you should get the matrix (the identity matrix).

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Part iii)

Take note that the dimensions of matrices A and B are 2x3 and 2x3 respectively. Since the inner dimensions do NOT match, this means that the matrix product AB is NOT defined. In other words, AB does NOT exist.

note: if A and B are mxn and nxp matrices, then the product AB is defined. It is only defined if the number of columns of A equals the number of rows of B. Otherwise, AB does NOT exist.

===========================================

Answer:

So the matrix product AB is not defined (ie it doesn't exist).

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Part iv)

I'm sure you meant to write correct? If so, then...

Step 1: Transpose matrix B to get . You do this by converting each row into a column (and vice versa).

AMP Rendering Error of 'B^T = (matrix(2,3,1,-1,4,2,3,5))^T = (matrix(3,2,1,2-1,3,4,5)) '.

Step 2: Multiply matrices and C. Since the inner dimensions are equal, this means that the matrix product is defined.

Since the first matrix is a 3 by 2 matrix and the second matrix is a 2 by 2 matrix, this means that the resulting matrix will be a 3 by 2 matrix.

So the final resulting matrix will look like:

note: the "x"s are just placeholders for now

--------------------------------------------------

Multiply the corresponding entries from the 1st row of the first matrix by the 1st column of the second matrix. After multiplying, add the values:

1st row, 1st column:

So the element in the 1st row, 1st column of the resulting matrix is . Now let's update the matrix:

--------------------------------------------------

Multiply the corresponding entries from the 1st row of the first matrix by the 2nd column of the second matrix. After multiplying, add the values:

1st row, 2nd column:

So the element in the 1st row, 2nd column of the resulting matrix is . Now let's update the matrix:

================================================================================

Multiply the corresponding entries from the 2nd row of the first matrix by the 1st column of the second matrix. After multiplying, add the values:

2nd row, 1st column:

So the element in the 2nd row, 1st column of the resulting matrix is . Now let's update the matrix:

--------------------------------------------------

Multiply the corresponding entries from the 2nd row of the first matrix by the 2nd column of the second matrix. After multiplying, add the values:

2nd row, 2nd column:

So the element in the 2nd row, 2nd column of the resulting matrix is . Now let's update the matrix:

================================================================================

Multiply the corresponding entries from the 3rd row of the first matrix by the 1st column of the second matrix. After multiplying, add the values:

3rd row, 1st column:

So the element in the 3rd row, 1st column of the resulting matrix is . Now let's update the matrix:

--------------------------------------------------

Multiply the corresponding entries from the 3rd row of the first matrix by the 2nd column of the second matrix. After multiplying, add the values:

3rd row, 2nd column:

So the element in the 3rd row, 2nd column of the resulting matrix is . Now let's update the matrix:

==============================================================================

Answer:

So AMP Rendering Error of 'B^T*C = (matrix(3,2,1,2,-1,3,4,5))*(matrix(3,2,2,-1,3,5)) = (matrix(3,2,8,9,7,16,23,21))'.

In other words,

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