SOLUTION: Can anyone help me with these problems please? 1) Use the matrix method to solve: 5x + 3y = 3 2x - 6y = 30 x = ____ y = ____ 2) Perform the following operations on

Algebra ->  Matrices-and-determiminant -> SOLUTION: Can anyone help me with these problems please? 1) Use the matrix method to solve: 5x + 3y = 3 2x - 6y = 30 x = ____ y = ____ 2) Perform the following operations on       Log On


   

Question 202458: Can anyone help me with these problems please?
1) Use the matrix method to solve:
5x + 3y = 3
2x - 6y = 30
x = ____
y = ____

2) Perform the following operations on matrices.
(4) + (8) = (____)
(7) (2) (____)

3) Perform the following operations on matrices.
(1 8)(7 6) = (___ ___)
(0 7)(7 4) (___ ___)

Thank you in advance for all your help.
~Sarah
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1

First, convert the system


system%285x+%2B+3y+=+3%2C2x+-+6y+=+30%29


into the matrix equation





So what we now need is inverse matrix of %28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29 to isolate %28matrix%282%2C1%2Cx%2Cy%29%29


Solved by pluggable solver: Finding the Inverse of a 2x2 Matrix

To find the inverse of the matrix A=%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29, we can follow these steps:

Step 1) Find the determinant



The determinant of %28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29 is abs%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29=-36. So this means that d=-36

Step 2) Swap the values



Now switch the highlighted values %28matrix%282%2C2%2Chighlight%285%29%2C3%2C2%2Chighlight%28-6%29%29%29 to get %28matrix%282%2C2%2Chighlight%28-6%29%2C3%2C2%2Chighlight%285%29%29%29

Step 3) Change the sign



Now change the sign of the highlighted values %28matrix%282%2C2%2C-6%2Chighlight%283%29%2Chighlight%282%29%2C5%29%29 to get %28matrix%282%2C2%2C-6%2Chighlight%28-3%29%2Chighlight%28-2%29%2C5%29%29

Step 4) Multiply by the inverse of the determinant



Multiply by 1%2Fd to get %281%2Fd%29%28matrix%282%2C2%2C-6%2C-3%2C-2%2C5%29%29

Plug in d=-36 to get %28-1%2F36%29%28matrix%282%2C2%2C-6%2C-3%2C-2%2C5%29%29

Step 5) Multiply -1%2F36 by every element in the matrix (simplify and reduce if possible)



Multiply -1%2F36 by EVERY element to get

Multiply to get %28matrix%282%2C2%2C-6%2F-36%2C-3%2F-36%2C-2%2F-36%2C5%2F-36%29%29

Reduce each element: %28matrix%282%2C2%2C1%2F6%2C1%2F12%2C1%2F18%2C-5%2F36%29%29


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


Answer:

So the inverse of %28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29 is %28matrix%282%2C2%2C1%2F6%2C1%2F12%2C1%2F18%2C-5%2F36%29%29

This means that if A=%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29 then A%5E%28-1%29=%28matrix%282%2C2%2C1%2F6%2C1%2F12%2C1%2F18%2C-5%2F36%29%29




Now left multiply both sides by the inverse matrix to get










%28matrix%282%2C1%2Cx%2Cy%29%29=%28matrix%282%2C2%2C3%2C-4%29%29


So this means that x=3 and y=-4


Note: let me know if you need help with matrix multiplication.





# 2

Simply add the corresponding components:





So





# 3


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

So the final resulting matrix will look like:


%28matrix%282%2C2%2Cx%2Cx%2Cx%2Cx%29%29


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:
%281%29%2A%287%29%2B%288%29%2A%287%29=7%2B56=63


So the element in the 1st row, 1st column of the resulting matrix is 63. Now let's update the matrix:

%28matrix%282%2C2%2C63%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:
%281%29%2A%286%29%2B%288%29%2A%284%29=6%2B32=38


So the element in the 1st row, 2nd column of the resulting matrix is 38. Now let's update the matrix:

%28matrix%282%2C2%2C63%2C38%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:
%280%29%2A%287%29%2B%287%29%2A%287%29=0%2B49=49


So the element in the 2nd row, 1st column of the resulting matrix is 49. Now let's update the matrix:

%28matrix%282%2C2%2C63%2C38%2C49%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:
%280%29%2A%286%29%2B%287%29%2A%284%29=0%2B28=28


So the element in the 2nd row, 2nd column of the resulting matrix is 28. Now let's update the matrix:

%28matrix%282%2C2%2C63%2C38%2C49%2C28%29%29








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


Answer:


So the solution is %28matrix%282%2C2%2C63%2C38%2C49%2C28%29%29

In other words,