BS-111 APPLIED Mathematics-I (credit: 4)

Determine the rank of the matrix A = [[1, 2, 3], [1, 2, 5], [2, 1, 3]]

State Cayley–Hamilton Theorem and use it to find the inverse of the matrix [[1, 1, 3], [1, 3, -3], [-2, -4, -4]]

Find the directional derivative of f(x,y,z) = xy² + yz² at the point (2, -1, 1) in the direction of vector i + 2j + 3k

Determine the rank of the matrix [[0, 1, -3, -1], [1, 0, 2, 0], [3, 1, 0, 2], [1, 1, -2, 0]]

Verify Cayley–Hamilton Theorem for matrix A = [[1, 0, 3], [2, 1, -1], [1, -1, 1]] and hence find A⁻¹