Trace-Inverse Gradient

Problem

For L(A) = trace(C A^{-1}) with constant matrix C, the gradient with respect to A is -(A^{-1} C A^{-1})^T.

Given the entries of the 2x2 matrix A as a, b, c, d and the entries of C as p, q, r, s, return the 2x2 gradient as a nested list. Return None when det(A) == 0.

Hint

Differentiate A A^{-1} = I to get d(A^{-1}) = -A^{-1} dA A^{-1}, then read off the gradient.

Test cases

case 1

in: [1,0,0,1,1,2,3,4]

expected: [[-1,-3],[-2,-4]]

case 2

in: [2,0,0,1,1,0,0,1]

expected: [[-0.25,0],[0,-1]]

case 3

in: [1,2,3,4,1,0,0,1]

expected: [[-5.5,3.75],[2.5,-1.75]]

case 4

in: [1,2,3,6,1,0,0,1]

expected: null

Starter codePython

def trace_inverse_gradient(a, b, c, d, p, q, r, s):
    # Return the 2x2 gradient
    # Your code here
    pass

4 test cases · run your code against them in the editor