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lecture_12 - MA 35100 LECTURE NOTES MONDAY FEBRUARY 8...

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MA 35100 LECTURE NOTES: MONDAY, FEBRUARY 8 Matrices which Commute We recap the main results from the previous lecture. Let A be an q × m matrix and B be an n × p matrix. a. The product B A is defined only when the number of rows of A (namely q ) is equal to the number of columns of B (namely q ). b. If p = q , then the n × m matrix C = B A is defined as the matrix of the linear transformation T ( ~x ) = B ( A ~x ). c. If the columns of A and ~v j , then the columns of B A and B ~v j . Explicitly, B A = B | | | ~v 1 ~v 2 · · · ~v m | | | = | | | B ~v 1 B ~v 2 · · · B ~v m | | | . As an example, we showed that 6 7 8 9 1 2 3 5 = 27 47 35 61 and 1 2 3 5 6 7 8 9 = 22 25 58 66 . This proves the following fact: Theorem. Matrix multiplication is noncommutative. Explicitly, let A and B be n × n matrices. Then A B and B A are both defined, but in general A B 6 = B A . This motivates a definition: Definition. Let A and B be n × n matrices. If A B = B A , then we say A and B commute .
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