Calculating the Determinant
Determinants of Special Matrices
Determinants of Manipulated Matrices
Invertibility
Cramer's Rule How to Examples Exercise
How to:
Cramer's Rule is a tool in helping solve linear systems, A*x = b, where we are given A, an NxN invertible (non-zero determinant) matrix, and b, an Nx1 column vector.
To calculate each component of x, that is, x[i, 1], we need to create a special matrix Ai.
Ai is the same as A with its i-th column replaced by b:
Ai = A
Ai[:, i] = b
Now we can compute x[i, 1] using our original matrix A and this new matrix Ai:
x[i, 1] = det(Ai) / det(A)
Examples:
Exercise: