In advanced engineering mathematics, and particularly in the field of differential equations, an understanding of Cramer's Rule is required to evaluate eigenvalue and eigenvector solutions. Cramer's Rule involves solving a system of linear equations by evaluating the determinants of matrices made up of co-efficients from the equations and the solution vector of those equations. Since most vector equations in practical application are 3-dimensional, determinants of 3 X 3 matrices are the most commonly evaluated. The procedure can easily be expanded to evaluate the determinant of a larger matrix.
Things You'll Need
Pen/Pencil
Paper
Calculator (optional)
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Instructions
1
Calculate the determinant of the bottom right 2 X 2 sub-matrix (D1). Multiply that value by the upper-left value in the matrix. These values are highlighted in Image 1.
D1 = 3*1 - 5*2 = 3 - 10 = -7
a11 * D1 = 2*(-7) = -14
2
Repeat this step for the top-center value and its associated 2 X 2 sub-matrix (D2).
D2 = 4*1 - 2*(-3) = 4 + 6 = 10
a12 * D2 = 1*10 = 10
3
Repeat this step for the top-right value and its associated 2 X 2 matrix (D3).
D3 = 4*5 - 3*(-3) = 20 + 9 = 29
a13 * D3 = 3 * 29 = 87
4
Use these 3 values to calculate the determinant for the full 3 X 3 matrix. The equation is:
D = a11*D1 - a12*D2 + a13* D3
D = -14 - 10 + 87 = 59