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Prove that A is non–singular, find A−1 and 7 9 product of elementary row matrices. ] 50 CHAPTER 2. MATRICES   1 2 k 1  4. Find the rational number k for which the matrix A =  3 −1 5 3 −5 is singular. ] 1 2 is singular and find a non–singular matrix −2 −4 P such that P A has last row zero. 5. Prove that A = 1 4 , verify that A2 − 2A + 13I2 = 0 and deduce that −3 1 1 = − 13 (A − 2I2 ). 6. If A = A−1   1 1 −1 1 . 7. Let A =  0 0 2 1 2 (i) Verify that A3 = 3A2 − 3A + I3 . (ii) Express A4 in terms of A2 , A and I3 and hence calculate A4 explicitly.

Then from AB = In we deduce (AB)B −1 = In B −1 and hence A = B −1 . The equation BB −1 = In then gives BA = In . Before we give the next example of the above criterion for non-singularity, we introduce an important matrix operation. 3 (The transpose of a matrix) Let A be an m × n matrix. Then At , the transpose of A, is the matrix obtained by interchanging the rows and columns of A. In other words if A = [aij ], then At ji = aij . Consequently At is n × m. The transpose operation has the following properties: 1.

1. 7 (Matrix product) Let A = [aij ] be a matrix of size m × n and B = [bjk ] be a matrix of size n × p; (that is the number of columns of A equals the number of rows of B). Then AB is the m × p matrix C = [cik ] whose (i, k)–th element is defined by the formula n cik = j=1 aij bjk = ai1 b1k + · · · + ain bnk . 2 1. 1 2 3 4 5 6 7 8 = 1×5+2×7 1×6+2×8 3×5+4×7 3×6+4×8 = 2. 5 6 7 8 1 2 3 4 = 23 34 31 46 ; 3. 1 2 4. 3 4 5. 1 −1 1 −1 3 4 = 3 4 6 8 1 2 = 11 1 −1 1 −1 = 1 2 3 4 5 6 7 8 19 22 43 50 ; ; ; = 0 0 .

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### (A, B)-Invariant Polyhedral Sets of Linear Discrete-Time Systems by Dorea C. E.

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