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

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) (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, ,

.
.
.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Let's compute the product of the matrices and

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:

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:

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

So the element in the 1st row, 3rd 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 2nd row of the first matrix by the 3rd column of the second matrix. After multiplying, add the values:
2nd row, 3rd column:

So the element in the 2nd row, 3rd 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:

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

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:

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

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

So this shows us that

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

Answer:

Since the product is NOT equal to the 3x3 identity matrix , this means that the two given matrices are NOT inverses of one another.
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