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lecture_7- Special Types of Matrices and Partitioned Matrices Part 2

Lecture_7- Special Types of Matrices and Partitioned Matrices Part 2

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MA 265 LECTURE NOTES: WEDNESDAY, JANUARY 23 Special Types of Matrices Linear Systems and Inverses. We return to the question of solving a system of linear systems: a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = b 2 . . . . . . . . . . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = b m Recall that we can express this system as a product of matrices: A = a 11 a 12 ... a 1 n a 21 a 22 ... a 2 n . . . . . . . . . . . . a m 1 a m 2 ... a mn , x = x 1 x 2 . . . x n and b = b 1 b 2 . . . b n = A x = b . For simplicity, assume that m = n . We will show that if A is a nonsingular matrix, then the system has a unique solution. Recall that we say that A is nonsingular (or invertible ) if there exists an n × n matrix A - 1 such that AA - 1 = A - 1 A = I n is the n × n identity matrix. First we show that the system has at least one solution x 1 . Indeed, denote x 1 = A - 1 b . Then we have A x 1 = A ( A - 1 b ) = ( AA - 1 ) b = I n b = b so that x 1 is a solution. Now we show that the system has at most one solution. Indeed, say that x 2 is another solution. Then we have x 2 = I n x 2 = ( A - 1 A ) x 2 = A - 1 ( A x 2 ) = A - 1 b = x 1 .
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