Let E1 Ek be n Times n elementary matrices and let A E1Ek

Let E_1, ...., E_k be n Times n elementary matrices and let A = E_1...E_k. Explain why Ax rightarrow = b rightarrow has a unique solution for all n Times 1 matrices b rightarrow. Given that A and B are n Times n symmetric matrices, Prove that A+ B is also symmetric.

Solution

13. We know that a n x n elementary matrix is a square obtained from the n × n identity matrix by an elementary row or column operation and that every elementary matrix is invertible and the inverse of an elementary matrix is an elementary matrix. Now, if A = E₁ E₂ … E_k , then A^-1 = (E₁ E₂ … E_k)^-1 = E_k^-1E_k-1^-1 … E₂^-1E₁^-1. Hence, if Ax = b, then x = A^-1 b =( E_k^-1E_k-1^-1 … E₂^-1E₁^-1)b is the quique solution of the equation Ax = b.

14. We know that a symmetric matrix is a square matrix which is equal to its transpose so that A^T=A and B^T = B. Also, (A+ B)^T = A^T + B^T = A + B. Therefore, A+B is also symmetric.