8. (30 marks) (a) Let S be an (m x m)-square matrix, and T be an (nx n)-square matrix. Define S Omxn U Onxm T Suppose S,

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8 30 Marks A Let S Be An M X M Square Matrix And T Be An Nx N Square Matrix Define S Omxn U Onxm T Suppose S 1 (30.09 KiB) Viewed 47 times

8. (30 marks) (a) Let S be an (m x m)-square matrix, and T be an (nx n)-square matrix. Define S Omxn U Onxm T Suppose S,T are both non-singular. Is U non-singular, or singular? Justify your answer with reference to the definition of non-singularity and singularity. (b) Let A be a (3 x 4)-matrix, B be a (3 x 5)-matrix, and C be a (4 x 5)-matrix. Let H be the (7 x 9)-matrix defined by A B H= - • Loaxa to 1. Suppose A is a reduced row-echelon form with 3 leading ones, and C is a reduced row-echelon form with 4 leading ones. What is the rank of H? Justify your answer. ii. Suppose A is of rank 3 and C is of rank 4. What is the rank of H? What is the nullity of H? Justify your answer. Remark. You may use, without proof, the result that two (px q)-matrices K, L are row-equivalent if and only if there is a non-singular (px p)-matrix J so that L= JK.