2'3The Inverse of a Matrix Study Book p 40,

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2.3 The Inverse of a Matrix Study Book p 40,
Larson, Ch 2.3

• Objectives be able to
• define the inverse of a square matrix
• find inverses by row-reduction
• write the inverse of a 2x2 matrix
• use A-1 (when it exists) to solve the linear
  system Ax b, or matrix eqn AX B

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• Square matrix A (n x n) is called
• invertible or non-singular
• if only if there exists matrix B (n x n)
• so that AB BA I.
• Then B is called A-1, the inverse of A.
• Example
• 2 1 is the inverse of 1 -1
• 1 1 -1 2
• ( vice versa) because their product gives I
  (Check!)

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• Note
• Only square matrices can be invertible
• and satisfy AB BA I
• Th 2.7 proves that if A-1 exists, it is unique
• ie a matrix cannot have 2 inverses!
• Finding the inverse of A
• means having to solve linear systems with
  coefficients A, and RHSs the columns of I.
• Example 1 2 X 1 0
• 3 5 0 1

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• We can do this by row-reducing A I
• If A reduces to I, the process succeeds,
• I reduces to A-1
• ie A I row- reduces to I A-1
• If A fails to reduce to I (ie a row of 0s
  appears) then A-1 does not exist.
• Examples (See 3 4, Larson p 69.)

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• In particular the 2 x 2 matrix
• A a b
• c d
• has inverse 1 d - b
• (ad - bc) - c a
• which exists if only if (ad - bc) is not
  0.
• So ad - bc determines if A is invertible or
  not.
• We call this number the determinant of A.
• Examples The determinants of the 2x2 matrices on
  slide 3 are 5 -6 -1 and 1-0 1,
  respectively.

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• Properties of Inverses follow see Th 2.8, 2.9,
  2.10
• Note the proofs of Th 2.9 2.10.
• System of linear equations, Ax b, can also be
  solved using A-1, if it exists, because
• Examples

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Homework

• Larson Edwards Ch 2.3.
• Master odd numbers, Q 1 - 35.
• Also do Ed 4 Q 37a-c, 42, 45, 48
• or Ed 5 Q 39a-c, 44, 47, 50.
• Write full solutions to
• Q 3, 13, 15, 25, 27(use tech),
• and Ed 4 Q 33, 34, 35, 36, 42, 45
• or Ed 5 Q 35, 36, 37, 38, 44, 47.

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Review Objectives

• Be able to
• define the inverse of a square matrix
• find inverses, when they exist, by row-reduction
• write down the inverse of a 2x2 matrix
• use A-1 (when it exists) to solve the system of
  linear eqns Ax b, or matrix eqn AX B.