2. [4 marks]. Let A be a given n x n matrix, where n is a...

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2. [4 marks]. Let A be a given n x n matrix, where n is a positive integer. Let S be the set of all n x n matrices B having the property that AB = BA. Let us prove that S is a subspace. (a) [2 marks]. Show that S is closed under vector addition. That is, suppose the matrices B and Care elements of S. Show that B + C must also be an element of S. (b) [2 marks]. Show that S is closed under scalar multiplication. That is, suppose that B is an element of S. Show that oB must also be an element of S for all o € R. 2. [4 marks]. Let A be a given n x n matrix, where n is a positive integer. Let S be the set of all n x n matrices B having the property that AB = BA. Let us prove that S is a subspace. (a) [2 marks]. Show that S is closed under vector addition. That is, suppose the matrices B and Care elements of S. Show that B + C must also be an element of S. (b) [2 marks]. Show that S is closed under scalar multiplication. That is, suppose that B is an element of S. Show that oB must also be an element of S for all o € R.

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