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2331_Notes_2o5_fill

2331_Notes_2o5_fill - Math 2331 Linear Algebra Sections 2.5...

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Math 2331 Linear Algebra Sections 2.5 Inverses of Matrices Recall the following basic facts of real number multiplication: The number 1 is called the multiplicative identity since 1* *1 x x x = = for all real numbers x. The numbers a and b are called multiplicative inverses if * 1 a b = In matrix multiplication the identity matrix is called I. Note that we use the generic term I to indicate an identity, you need the context to determine the size of the square matrix I. For all matrices A * I = A m n × m n n n × × I * A = A m m × m n m n × × Two square matrices A and B are called inverses of each other if AB = BA = I Given the matrix A, its inverse is denoted 1 A Properties of Inverses 1. A square n x n matrix has an inverse EXACTLY when it has a full set of n pivots. Notice that when solving Ax = b, it had a unique solution when A had a full set of pivots. If A has an inverse, 1 1 A Ax A b = If A doesn't have a full set of pivots, we can find a nonzero solution to Ax = 0 Suppose A had an inverse but not a full set of pivots Ax = 0

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