SOLUTION: can someone please help: Determine whether the two matrices are inverses of each other by computing their product. 1. 2, -1, 0 -1, 1, -2 1, 0, -1 2. 1, -1, 2 -3, -2

Algebra ->  Matrices-and-determiminant -> SOLUTION: can someone please help: Determine whether the two matrices are inverses of each other by computing their product. 1. 2, -1, 0 -1, 1, -2 1, 0, -1 2. 1, -1, 2 -3, -2      Log On


   

Question 166219: can someone please help:
Determine whether the two matrices are inverses of each other by computing their product.
1.
2, -1, 0
-1, 1, -2
1, 0, -1
2.
1, -1, 2
-3, -2, 4
-1, 1, 1
yes or no
Found 2 solutions by Fombitz, jim_thompson5910:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Multiply rows of the first matrix by columns of the second matrix to get the product.
Example: First entry of the product matrix, P%5B1%2C1%5D,
P%5B1%2C1%5D=2%281%29%2B%28-1%29%28-3%29%2B0%28-1%29=2%2B3%2B0=5
P%5B1%2C2%5D=2%28-1%29%2B%28-1%29%28-2%29%2B0%281%29=-2%2B2%2B0=0
P%5B1%2C3%5D=2%282%29%2B%28-1%29%284%29%2B0%281%29=4-4%2B0=0
.
.
.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's compute the product of the matrices %28matrix%283%2C3%2C2%2C-1%2C0%2C-1%2C1%2C-2%2C1%2C0%2C-1%29%29 and %28matrix%283%2C3%2C1%2C-1%2C2%2C-3%2C-2%2C4%2C-1%2C1%2C1%29%29


Since the first matrix is a 3 by 3 matrix and the second matrix is a 3 by 3 matrix, this means that the resulting matrix will be a 3 by 3 matrix (just look at the outer dimensions).

So we have this matrix set up:


%28matrix%283%2C3%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%29%29


note: the "x"s are just placeholders for now. When we find the answers, we'll fill in the "x"s with those answers.


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

To multiply the matrices, we need to follow these steps:



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:
%282%29%2A%281%29%2B%28-1%29%2A%28-3%29%2B%280%29%2A%28-1%29=2%2B3%2B0=5


So the element in the 1st row, 1st column of the resulting matrix is 5. Now let's update the matrix:

%28matrix%283%2C3%2C5%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%29%29
--------------------------------------------------




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:
%282%29%2A%28-1%29%2B%28-1%29%2A%28-2%29%2B%280%29%2A%281%29=-2%2B2%2B0=0


So the element in the 1st row, 2nd column of the resulting matrix is 0. Now let's update the matrix:

%28matrix%283%2C3%2C5%2C0%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%29%29
--------------------------------------------------




Multiply the corresponding entries from the 1st row of the first matrix by the 3rd column of the second matrix. After multiplying, add the values:
1st row, 3rd column:
%282%29%2A%282%29%2B%28-1%29%2A%284%29%2B%280%29%2A%281%29=4%2B-4%2B0=0


So the element in the 1st row, 3rd column of the resulting matrix is 0. Now let's update the matrix:

%28matrix%283%2C3%2C5%2C0%2C0%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%29%29
--------------------------------------------------


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




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:
%28-1%29%2A%281%29%2B%281%29%2A%28-3%29%2B%28-2%29%2A%28-1%29=-1%2B-3%2B2=-2


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

%28matrix%283%2C3%2C5%2C0%2C0%2C-2%2Cx%2Cx%2Cx%2Cx%2Cx%29%29
--------------------------------------------------




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:
%28-1%29%2A%28-1%29%2B%281%29%2A%28-2%29%2B%28-2%29%2A%281%29=1%2B-2%2B-2=-3


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

%28matrix%283%2C3%2C5%2C0%2C0%2C-2%2C-3%2Cx%2Cx%2Cx%2Cx%29%29
--------------------------------------------------




Multiply the corresponding entries from the 2nd row of the first matrix by the 3rd column of the second matrix. After multiplying, add the values:
2nd row, 3rd column:
%28-1%29%2A%282%29%2B%281%29%2A%284%29%2B%28-2%29%2A%281%29=-2%2B4%2B-2=0


So the element in the 2nd row, 3rd column of the resulting matrix is 0. Now let's update the matrix:

%28matrix%283%2C3%2C5%2C0%2C0%2C-2%2C-3%2C0%2Cx%2Cx%2Cx%29%29
--------------------------------------------------


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




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:
%281%29%2A%281%29%2B%280%29%2A%28-3%29%2B%28-1%29%2A%28-1%29=1%2B0%2B1=2


So the element in the 3rd row, 1st column of the resulting matrix is 2. Now let's update the matrix:

%28matrix%283%2C3%2C5%2C0%2C0%2C-2%2C-3%2C0%2C2%2Cx%2Cx%29%29
--------------------------------------------------




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:
%281%29%2A%28-1%29%2B%280%29%2A%28-2%29%2B%28-1%29%2A%281%29=-1%2B0%2B-1=-2


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

%28matrix%283%2C3%2C5%2C0%2C0%2C-2%2C-3%2C0%2C2%2C-2%2Cx%29%29
--------------------------------------------------




Multiply the corresponding entries from the 3rd row of the first matrix by the 3rd column of the second matrix. After multiplying, add the values:
3rd row, 3rd column:
%281%29%2A%282%29%2B%280%29%2A%284%29%2B%28-1%29%2A%281%29=2%2B0%2B-1=1


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

%28matrix%283%2C3%2C5%2C0%2C0%2C-2%2C-3%2C0%2C2%2C-2%2C1%29%29
--------------------------------------------------






So this shows us that




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

Answer:


Since the product %28matrix%283%2C3%2C5%2C0%2C0%2C-2%2C-3%2C0%2C2%2C-2%2C1%29%29 is NOT equal to the 3x3 identity matrix %28matrix%283%2C3%2C1%2C0%2C0%2C0%2C1%2C0%2C0%2C0%2C1%29%29, this means that the two given matrices are NOT inverses of one another.