solve m using Cramers rule

Question

m-n+r+3=0

2n+4-m=r

m(over 2 )-1=r( over 3)

 

solve m by using cramer's rule

Answer 1

Cramer's rule is a long and involved process:

 

m-n+r+3 = 0  →      m-n+r = -3                  →        m-n + r = -3

2n+4-m = r    →      -m+2n-r=-4                 →       -m+2n-r=-4

m/2 -1 = r/3  →    (1/2)m+0n-(1/3)r = 1     →     3m+0n-2r=6    

 

I rearranged the equations to the form am + bn + cr = d 

Then I got rid of fractions by multiplying the bottom equation by 6

 

I can now write the matrix using the coefficients...  Please pretend there are long straight lines outlining

each matrix...

 

      matrix D      Ans. Matrix

     1   -1    1            -3

    -1   2   -1              4

     3   0    -2             6 

 

We need to find the determinant of Matrix D...

1)Rewrite the first 2 columns on the right end

2)Multiply the downward diagonals and add that

3)Multiply the upward diagonals.. add that...

    subtract from the sum of the upward diagonals

  

   1    -1    1    1   -1       1(2)(-2)+(-1)(-1)(3)+(1)(-1)(0)=-4+3=-1

  -1     2   -1   -1    2       3(2)(1)+(0)(1)(1)+(-2)(-1)(-1)=6-2=4

   3     0   -2    3    0                                   -1-4 = -5  = Det(D)

 

Now we need to find matrix D_m and Det(D_m)

 

Replace the 1st column of D (m coefficients) with values from answer matrix

Then find determinant as above

 

  - 3    -1    1   -3    -1     (-3)(2)(-2)+(-1)(-1)(6)+(1)(-4(0)=12+6+0=18

  -4      2   -1   -4     2     (6)(2)(1)+(0)(-1)(-3)+(-2)(-4)(-1)=12+0-8 =4

   6      0   -2    6     0                                     Det(D_m) = 18-4 = 14

 

Now find D_n and Det(D_n)

 

Replace 2nd column of D (n coefficients) with answer matrix

and find determinant

 

 1   -3   1   1   -3       (1)(-4)(-2)+(-3)(-1)(3)+(1)(-1)(6)=8+9-6=11

-1   -4  -1  -1  -4       (3)(-4)(1)+(6)(-1)(1)+(-2)(-1)(-3)=-12-6-6=-24

 3    6   -2   3   6                                       Det(D_n) = 11-(-24) = 35

 

Find D_r and Det(D_r)

Replace the 3rd column of D (r coefficients) with answer matrix

and find determinant

 

 1   -1   -3   1   -1      (1)(2)(6)+(-1)(-4)(3)+(-3)(-1)(0)=12+12+0=24

-1    2   -4  -1    2      (3)(2)(-3)+(0)(-4)(1)+(6)(-1(-1)=-18+0+6=-12

 3    0     6   3    0                                          Det(D_r)=24-(-12)=36

 

Cramers rule says   m = D_m/D = -14/5

                             n = D_n/D = 35/(-5) = -7

                             r = D_r/D = 36/(-5)

 

m = 14/(-5)=-14/5    n = -7    r = -36/5

 

You can verify by plugging back into the original equations to verify the answer.

 

I used the 1st equation m-n+r=-3

 

-14/5 +7 -36/5 = -3

-50/5 + 7 = -3

-10 + 7 = -3

-3 = -3

The answer works