MAML, Matrices, Determinants and Systems of Equations: 4-by-4 Matrices
This is a MAML (Maine Association of Math Leagues) problem from Month (Meet Num of Year), Round Num, Problem Num: http://maml.net/problems.htm
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Condensed Writing Solution:
We must find the determinant of the following matrix:
|1 2 3 0|
|2 1 0 0|
|2 0 1 5|
|1 1 1 1|
To do this, we need to find the determinant of each minor 3x3 matrix and multiply the determinant by the excluded element and the appropriate sign (https://people.richland.edu/james/lec.... Let's say the determinant of each minor matrix is d_i. In this case, the determinant of the whole determinant will be:
1*d_1-2*d_2+3*d_3-0*d_4
We can get rid of the last term, so there is no need to calculate d_4:
d_1-2*d_2+3*d_3
Now, we can find the determinant of each minor matrix using the diagonal method of solving 3x3 matrices (http://demonstrations.wolfram.com/33D...
d1=(1+0+0)-(0+5+0)=-4
1 0 0| 1 0
0 1 5| 0 1
1 1 1| 1 1
d2=(2+0+0)-(0+10+0)=-8
2 0 0 | 2 0
2 1 5 | 2 1
1 1 1 | 1 1
d3=(0+5+0)-(0+10+2)=-7
2 1 0 | 2 1
2 0 5 | 2 0
1 1 1 | 1 1
Now, we need to plug these variables into our original expression and evaluate:
d_1-2*d_2+3*d_3=-4-2*-8+3*-7=-4+16-21=-9 ANSWER