Exercise 1

Consider the following matrices:

$$ A = \begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix},\quad B = \begin{pmatrix} -4 & 2 \\ 3 & 1 \end{pmatrix},\quad C = \begin{pmatrix} 4 & 5 & 6 \\ 8 & -1 & 0 \\ 3 & 4 & 0 \end{pmatrix},\quad D = \begin{pmatrix} 2 & 4 & -1 \\ 5 & -3 & 7 \\ 1 & 4 & 2 \end{pmatrix} $$

Questions:

1. Compute 2C − D, CD, C², tA, tC, Tr(B), Tr(D).
2. Compute det(A) and det(D).
3. Compute AB and deduce A⁻¹.

Exercise 2

Let A be a square matrix such that:

A³ = 3A² - 5A + 2I

where I is the identity matrix. Show that A is invertible and find its inverse A^-1.