Lab 11 · Lecture 11 · Linear Systems, Cramer's Rule and the Inverse
Cramer and Inverse Lab
Solve your own systems by Cramer's rule and the adjugate inverse, measure conditioning, and see how fast the cost grows.
Type any 2×2, 3×3 or 4×4 system as an augmented matrix [A | b]. The tool shows D, every Dᵢ with the replaced column shaded, and the solution: use it to check your hand calculations.
Switch "Show" between the adjugate inverse and the norms. Try the nearly parallel system and compare κ with the circuit.
Ill-conditioning in pictures
Operation counts on a log scale. Move n and compare Cramer with elimination.
Cramer, inverse and κ
Try this
Cost of Cramer's rule
Try this
Q1 Solve x+y+z=6, 2x−y+z=3, x+2y−z=2 by Cramer's rule.
D=1(1−2)−1(−2−1)+1(4+1)=7. D1=7, D2=14, D3=21. So x=1, y=2, z=3.
Q3 Find ∥v∥1, ∥v∥2, ∥v∥∞ for v=(3,−4,1), and ∥A∥1, ∥A∥∞ for A=(13−24).
8, 26=5.099, 4. Column sums 4 and 6: ∥A∥1=6. Row sums 3 and 7: ∥A∥∞=7.
Q4 Why is Cramer's rule not used for a system of 20 equations?
It needs 21 determinants of size 20; by cofactor expansion that is about 1.4×1020 operations (about 4400 years at 109 per second). Gaussian elimination needs about 32203≈5900 operations.
Q5 A matrix has κ(A)=106 and the data b are correct to 8 significant digits. How many digits of x can you trust?