§7.2, Exercise 8. Let A be an n × n matrix. Suppose that A² + A+ In = 0. Prove that A is invertible.

2. Prove that if A is a symmetric positive definite matrix then its inverse, A-¹, is also symmetric and positive definite.

Let A be a 4 × 4 matrix and let x1, x2, X3, X4 be given by -183 -46 139 49 -93 -23 71 25 X1 = X2 = X3 = X4 = 112 28 -85 -30 4 1 -3 -1 If 1 1 Ax1 1 Ax2 = Ax3 Ax4 1 1 1 1 show that A is invertible and find A-1.

Let A be an invertible 3x3 matrix. Mark all that are necessarily true about this matrix: The dimension of the vector space spanned by the columns of A is at most 2. The columns of A are linearly independent. O A is diagonalizable O O is not an eigenvalue of A.