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# 2.8 - Section 2.8 Inverses and Determinants Theorem If A is...

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Section 2.8 Inverses and Determinants Theorem : If A is invertible, then A -1 is unique. Theorem : If A is an nXn invertible matrix, then (a) A -1 is also invertible and (A -1 ) -1 = A Why ? Because by definition, AA -1 = I n and A -1 A = I n . (b) If B is also invertible and nXn, then AB is invertible and (AB) -1 = B -1 A -1 Why ? Because (AB) (B -1 A -1 ) = A A -1 = I n

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(c) If k is a nonzero scalar then kA is invertible and (kA) -1 = 1 1 A k Example: If 11 10 0 1 2 0 210 3 3 1 54 1 6 7 2 Aa n d B −−   =− = − −  , find (AB) -1 . 1 () AB B A == 12 0 1 0 0 3 3 1 67 2 5 4 1   −−−  52 0 411 10 1 2 Theorem: det(AB) = det(A)det(B) Example: Suppose det(A) = 2 and

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2.8 - Section 2.8 Inverses and Determinants Theorem If A is...

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