SOLUTION: Solve the system of equations by using the inverse of the coefficient matrix. X+2y=4 2x-3y=1

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Question 567404: Solve the system of equations by using the inverse of the coefficient matrix.
X+2y=4
2x-3y=1
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
system%28x%2B2y=4%2C%0D%0A2x-3y=1%29

Write as the 2x2 coefficient matrix multiplied by the 2x1 variable 
matrix and set it equal to the 2x1 constant matrix:

%28matrix%282%2C2%2C1%2C2%2C2%2C-3%29%29%28matrix%282%2C1%2Cx%2Cy%29%29 = %28matrix%282%2C1%2C4%2C1%29%29

To find the inverse of the 2x2 coefficient matrix: %28matrix%282%2C2%2C1%2C2%2C2%2C-3%29%29

1. Swap the upper left and lower right emements: %28matrix%282%2C2%2C-3%2C2%2C2%2C1%29%29

2. Change the signs of the upper right and lower left elements: %28matrix%282%2C2%2C-3%2C-2%2C-2%2C1%29%29

3. Calculate the determinant, either of the original matrix or the one that is the result of step 3 for their determinants are the same: 
   
original matrix's determinant:  (1)(-3)-(2)(2) = -3-4 = -7
step 3 matrix;s determinant:  (-3)(1) - (-2)(-2) = -3-4 = -7

4. Divide each term in the matrix by the value of the determinant:

%28matrix%282%2C2%2C3%2F7%2C2%2F7%2C2%2F7%2C-1%2F7%29%29

Left-multiply both sides of the matrix equation by this inverse:

%28matrix%282%2C2%2C3%2F7%2C2%2F7%2C2%2F7%2C-1%2F7%29%29%28matrix%282%2C2%2C1%2C2%2C2%2C-3%29%29%28matrix%282%2C1%2Cx%2Cy%29%29 = %28matrix%282%2C2%2C3%2F7%2C2%2F7%2C2%2F7%2C-1%2F7%29%29%28matrix%282%2C1%2C4%2C1%29%29

%28matrix%282%2C1%2Cx%2Cy%29%29 = 

%28matrix%282%2C1%2Cx%2Cy%29%29 = %28matrix%282%2C1%2C12%2F7%2B2%2F7%2C8%2F7-1%2F7%29%29

%28matrix%282%2C4%2C7%2F7%2C%22%22%2C%22%22%2C0%2F7%2C0%2F7%2C%22%22%2C%22%22%2C7%2F7%29%29%28matrix%282%2C1%2Cx%2Cy%29%29 = %28matrix%282%2C1%2C14%2F7%2C7%2F7%29%29

%28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29%28matrix%282%2C1%2Cx%2Cy%29%29 = %28matrix%282%2C1%2C2%2C1%29%29

%28matrix%282%2C1%2C1x%2B0y%2C0x%2B1y%29%29 = %28matrix%282%2C1%2C2%2C1%29%29

%28matrix%282%2C1%2Cx%2Cy%29%29 = %28matrix%282%2C1%2C2%2C1%29%29

x=2, y=1

Edwin