Step-by-step explanation:
2a.
the determinant of a 4×4 matrix
a b c d
e f g h
i j k l
m n o p
is
a×det(fgh to nop) - b×det(egh to mop) +
c×det(efh to mnp) - d×det(efg to mno)
the determinant of a 3×3 matrix e.g.
f g h
j k l
n o p
is
f×det(kl to op) - g×det(jl to np) + h×det(jk to no)
the determinant of a 2×2 matrix e.g.
k l
o p
is
k×p - l×o
so, for A that means
1 1 1 1
1 2 4 2
0 1 0 0
2 19 2 1
1×det(2 4 2 to 19 2 1) - 1×det(1 4 2 to 2 2 1) +
1×det(1 2 2 to 2 19 1) - 1×det(1 2 4 to 2 19 2)
det(2 4 2 to 19 2 1) =
2×det(0 0 to 2 1) - 4×det(1 0 to 19 1) + 2×det(1 0 to 19 2) =
= 2×(0×1 - 0×2) - 4×(1×1 - 0×19) + 2×(1×2 - 0×19) =
= 2×0 - 4×1 + 2×2 = -4 + 4 = 0
det(1 4 2 to 2 2 1) =
1×det(0 0 to 2 1) - 4×det(0 0 to 2 1) + 2×det(0 0 to 2 2) =
= 1×0 - 4×0 + 2×0 = 0
det(1 2 2 to 2 19 1) =
1×det(1 0 to 19 1) - 2×det(0 0 to 2 1) + 2×det(0 1 to 2 19) =
= 1×(1×1 - 0×19) - 2×0 + 2×(0×19 - 1×2) =
= 1×1 + 2×-2 = 1 - 4 = -3
det(1 2 4 to 2 19 2) =
1×det(1 0 to 19 2) - 2×det(0 0 to 2 2) + 4×det(0 1 to 2 19) =
= 1×(1×2 - 0×19) - 2×0 + 4×(0×19 - 1×2) =
= 1×2 + 4×-2 = 2 - 8 = -6
det(A) = 1×0 - 1×0 + 1×-3 - 1×-6 = -3 + 6 = 3
2b.
x =
x1
x2
x3
x4
Ax = b means
1×x1 + 1×x2 + 1×x3 + 1×x4 = 2
1×x1 + 2×x2 + 4×x3 + 2×x4 = 4
0×x1 + 1×x2 + 0×x3 + 0×x4 = 2 => x2 = 2
2×x1 + 19x2 + 2×x3 + 1×x4 = 1
Cramer's rule says
x2 = det(A2)/det(A) = det(A2)/3
A2 means replace column 2 of A by b :
1 2 1 1
1 4 4 2
0 2 0 0
2 1 2 1
that is way more complicated than just using equation 3 up there. please use a matrix calculator.
yes, as expected, det(A2) = 6
x2 = det(A2)/3 = 6/3 = 2