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Section 2.2 Matrix Inverse In its most basic form a matrix A has an inverse if there is a matrix B such that AB BA I and if this matrix B exists at all then we label it B A −1 Theorem 4 hints at a future method to determine if a matrix has an inverse or not. There is a function called the determinant that you apply it to a square matrix and the function spits out a number. If the collection of matrices are matrices with only real numbers then the output number is a real numbers and if we allow complex numbers into our matrix then the output number is a complex number. det : Collection of Square Matrices → Either or ℂ For a 2x2 matrix a b det c d 5 3 det 2 4 ad − cb 5∗4−2∗3 20 − 6 14 det 2 3i i 2i 4 2 3i4 − 2i ∗ i 8 12i − 2i 2 10 12i Important Fact 1 1 A has an inverse or is called invertible if detA ≠ 0 Important Fact 2 If A is a n x n matrix and A is invertible then for each b in the range of linear transformation T there is a unique solution to Ax b A −1 Ax A −1 b x A −1 b If in the back of your mind you are thinking doesn’t this mean that the transformation T is one - to - one they you would be correct. Example 3, page 104 This is a very interesting example for the Civil Engineers and the Material Scientists in terms of the "stiffness matrix" and the deflection vector. Example 2, page 109 A 3 2 8 5 detA 35 − 82 15 − 16 −1 Since the determinant is not zero matrix A does have an inverse. 2 1 detA A −1 −1 ∗ 5 −2 −8 3 5 −2 −8 3 −5 2 8 −3 and check A −1 A −5 2 3 2 8 −3 8 5 1 0 0 1 I2 Example 6, page 109 Use the inverse to solve 7x 1 3x 2 −9 − 6x 1 − 3x 2 4 A 7 3 −6 −3 , b −9 4 Find the inverse A −1 1 1 −2 − 73 Get the solution 3 x A −1 b 1 −9 1 −2 − −5 26 3 7 3 4 x1 x2 Example 12, page 109 Use matrix algebra to show that if A is invertible and D satisfies AD I then D A −1 Proof AD I A −1 AD A −1 I A −1 AD A −1 ID A −1 D A −1 Example 18, page 110 Solve the matrix equation AB BC for A assuming that A, B,and C are all invertible. Solution AB BC ABB −1 BCB −1 A BCB −1 4