Matrix Inverses The Inverse of a Matrix Defn. The inverse of a - - PowerPoint PPT Presentation

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Aug 18, 2022 544 likes •662 views

Matrix Inverses The Inverse of a Matrix Defn. The inverse of a square matrix A , de- noted A 1 , is the matrix such that AA 1 = A 1 A = I . Defn. The inverse is not guaranteed to exist. If it exists, then A is invertible ; otherwise

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Matrix Inverses

SLIDE 2

The Inverse of a Matrix

Defn. The inverse of a square matrix A, de- noted A−1, is the matrix such that AA−1 = A−1A = I. Defn. The inverse is not guaranteed to exist. If it exists, then A is invertible; otherwise A is not invertible or singular.

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Matrix Equation with Invertible Matrix

Fact. If matrix A is invertible, then Ax = b has unique solution x = A−1b.

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Inverse of a 2 × 2 Matrix

The inverse of a 2 × 2 matrix has formula: a b c d −1 = 1 ad − bc d −b −c a

The formula also captures when the inverse ex-

ists: the matrix is invertible if and only if ad − bc = 0.

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Obtaining Matrix Inverses by Reduction

One way to find the inverse is to solve the collec- tion of n vector equations Ax = e1, . . . , Ax = en (where the ej are the columns of In as before). Equivalently: ALGOR To find inverse of matrix A, augment with the identity matrix In, then bring to re- duced row echelon form.

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Example Inverse Calculation

C = 3 −5 −5 9

is augmented to

3 −5 1 0 −5 9 0 1

This reduces to

1 0 9/2 5/2 0 1 5/2 3/2

so that

C−1 = 9/2 5/2 5/2 3/2

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Formulas

Fact. If A and B are square matrices of the same size: (a) (A−1)−1 = A (b) (AB)−1 = B−1A−1 (Note the reversal!) (c) (AT)−1 = (A−1)T.

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Characterization of Invertible Matrices

The big theorem. Fact. For an n × n matrix A, the following are equivalent: ≻ A is invertible ≻ A has n pivots ≻ A is row equivalent to In ≻ Ax = 0 has a unique solution ≻ the columns of A are linearly independent ≻ the columns of A span Rn ≻ the range of transform x → Ax is all of Rn

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Summary

The inverse of a square matrix A is the matrix A−1 such that their product is the identity. If inverse exists, then A is invertible; otherwise A is singular. If matrix A is invertible, then Ax = b has unique solution A−1b. The inverse of a 2×2 matrix has formula

1 ad−bc

−b −c a

One way to find the inverse is to augment with

the identity matrix and bring to reduced row echelon form.

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Summary (cont)

An n × n matrix is invertible whenever it has n pivots; equivalently the columns are linearly in- dependent and span Rn.

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