# 2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A...

###### Question:

2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M E Mn,n(R) such that M € V. You may find it useful to use (a), along with a proof by contradiction. (C) (5 marks) Suppose T : Mn,n(R) → W is linear, and T(A)  0w for some nxn matrix A. Show that there exists an invertible matrix M such that T(M) + 0w.

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