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Each of the archetypes below is a system of equations with a square coefficient matrix, or is a square matrix itself. Compute the determinant of each matrix, noting how Theorem SMZD indicates when the matrix is singular or nonsingular.
Archetype A, Archetype B, Archetype F, Archetype K, Archetype L
A <- matrix(c(1,2,1,-1,1,1,2,1,0), nrow=3)
A
## [,1] [,2] [,3]
## [1,] 1 -1 2
## [2,] 2 1 1
## [3,] 1 1 0
det(A)
## [1] 0
det A=0, per Theorem SMZD A is singular.
Archtype B
B <- matrix(c(-7,5,1,-6,5,0,-12,7,4), nrow=3)
det(B)
## [1] -2
det B≠0, per Theorem SMZD B is nonsingular.
Archtype F
F<-matrix(c(33,99,78,-9,-16,-47,-36,2,10,27,17,3,-2,-7,-6,4), nrow=4)
det(F)
## [1] -18
det F≠0, per Theorem SMZD F is nonsingular.
Archtype K
K <- matrix(c(10,12,-30,27,18,18,-2,-21,30,24,24,-6,-23,36,30,24,0,-30,37,30,-12,-18,39,-30,-20), nrow=5)
det(K)
## [1] 16
det K≠0, per Theorem SMZD K is nonsingular.
Archtype L
L <- matrix(c(-2,-6,10,-7,-4,-1,-5,7,-5,-3,-2,-4,7,-6,-4,-4,-4,10,-9,-6,4,6,-13,10,6), nrow=5)
det(L)
## [1] -4.437343e-30
As L is closer to 0, inversibility is going to be checked.