Inverse Matrices

﹏ヽ暗。殇╰゛Y 2022-05-11 13:16 156阅读 0赞

### 文章目录 ###

*   *  DEFINITION
     *  IMPORTANT NOTES
     *  The Inverse of Product AB
     *  Calculating $A^\{-1\}$ by Gauss-Jordan Elimination

## DEFINITION ##

*   The matrix A is **invertible** if there exists a matrix  A − 1 A^\{-1\} A−1 that allows  
     (1) A − 1 A = I    a n d    A A − 1 = I A^\{-1\}A=I\\space \\space and\\space\\space AA^\{-1\}=I\\tag\{1\} A−1A=I  and  AA−1=I(1)

## IMPORTANT NOTES ##

1. The inverse exists if and only if elimination produces n pivots (n by n matrix)

2. The matrix A cannot have two different inverses

3. If A is invertible, the one and only solution to Ax=b is x= A − 1 A^\{-1\} A−1b

4. Suppose there is a nonezero vector x to get Ax=**0**. Then A cannot have an inverse.  
  if A is invertible ,then Ax=**0** can only have the zero solution x= A − 1 A^\{-1\} A−1 **0**=**0**

5.A diagonal matrix is invertible provided there is no diagonal entries are zero

If  A = \{ d 1 ⋱   ⋱ d n \} A= \\left\\\{ \\begin\{matrix\} d\_1 & & & \\\\ &\\ddots & \\ & \\\\ & & \\ddots & \\\\ & & & d\_n \\end\{matrix\} \\right\\\} A=⎩⎪⎪⎨⎪⎪⎧d1⋱ ⋱dn⎭⎪⎪⎬⎪⎪⎫     then      A − 1 A^\{-1\} A−1=  \{ 1 d 1 ⋱   ⋱ 1 d n \} \\left\\\{ \\begin\{matrix\} \\frac\{1\}\{d\_1\} & & & \\\\ &\\ddots & \\ & \\\\ & & \\ddots & \\\\ & & & \\frac\{1\}\{d\_n\} \\end\{matrix\} \\right\\\} ⎩⎪⎪⎨⎪⎪⎧d11⋱ ⋱dn1⎭⎪⎪⎬⎪⎪⎫

## The Inverse of Product AB ##

*   If A and B are invertible then so is AB. Inverse of a product AB is  
     ( A B ) − 1 = B − 1 A − 1 (AB)^\{-1\}=B^\{-1\}A^\{-1\} (AB)−1=B−1A−1  
    NOTE: the order is reversed

## Calculating  A − 1 A^\{-1\} A−1 by Gauss-Jordan Elimination ##

*   Multiply \[ A   I A\\space I A I\] by  A − 1 A^\{-1\} A−1 to get \[ I   A − 1 I \\space A^\{-1\} I A−1\]