Problem 2 - Numpy Find the determinant of the given matrix, A: 2 3 -64 0 4 1 8 0 -21 7 3 7 6 -1 2 5 9

display the coefficient matrix A and the augmented matrix B for the given system.  x1 − x2 = −1 x1 + x2 = 3

(Cayley-IHamilton Theorem) The Cayley-Ilamilton theorem is a powerful theorem in Linear algebra that states: every square matrix of real numbers satisfies its own char- acteristic equation. For 2 x 2 matrices this can be seen in the following way. First, we introduce the trace of the matrix tr A 41,1 +22 and the determinant det A = a11a22 - 41,22,1- Then, the chacteristic polynomial is the quadratic given by p(A) = x – (tr A)A + (det. A). Then, for any 2 x 2 matrix, if you plug it into this polynomial, you should always get a zero matrix. c_h_test Function: Input parameters: • a single 2 x2 matrix for which you would like to test the Cayley Ilamilton Theorem Local Variables: • two scalars to represent the trace and determinant Output parameters: a single 2 x2 which is the result of evaluating the charcteristic polynomial at A (note: think carefully about the constant term) A possible sample case is: > mat_B = c_h_test([3, 2 ; 2, 0]) mat B = 0 0

) Assume A is k × n-matrix and Q is n x n-invertible matrix. Prove that rank(AQ) = rank(A).