In all this problem, the matrices are square. 1- Prove that the set of squares matrices M, (R) is an algebra with dimension n and gives a basis of it. 2- Prove that the subset of invertible matrices of M (R) is closed by multiplication but not closed by addition. Deduce that this subset is not a subspace. 3- Let D E M (R) be a diagonal and invertible matrix. Give directly its inverse D-. 4- If A is a triangular and invertible matrix, how to find its inverse A-? 5- Prove that a product or a sum of lower triangular matrices is a lower triangular matrix. Using the transpose, deduce the same property for upper triangular matrices. 6- Prove that the set of lower triangular matrices is a subalgebra of M, (R) and find its dimension. Deduce the same property for the set of upper triangular matrices. 7- Prove the equality: vx,y ER, (Ax, y) = (x, AT y), where A is the transpose of A. 8- A symmetric matrix is a matrix satisfying A = AT. Prove that the subset of the symmetric matrices is a subspace of M, (R). Give its dimension. 9- Prove that (AB)T = BT AT. 10-Prove that the inverse of the transpose is the transpose of the inverse: (A)- = (A-) 11- Prove that a positive definite matrix is invertible. 12- Let A M (R) be a positive definite matrix. Prove the following: a) a > 0, Vi= 1,..., n. b) max|a|= maxlaul.