11.(a) Prove: If A is invertible, then det(A-) =1det(A)(b) Prove: If A is inverible, then adj(A) is invertible and[adj(A)]-det(A)1A = adj(A¯). NOT allowed to use CLOType here to search

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=) Prove: If A is invertible, then det(A-1) = ) Prove: If A is inverible, then adj(A) is inve

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

11.(a) Prove: If A is invertible, then det(A-) =
1
det(A)
(b) Prove: If A is inverible, then adj(A) is invertible and
[adj(A)]-
det(A)
1
A = adj(A¯). NOT allowed to use C
L
OType here to search

Expert Solution

Step 1

Introduction:

In the case of real numbers, the inverse of any real number a was the number $a^{- 1}$ , so that a multiplied by $a^{- 1}$ equaled 1. We knew that the inverse of a real number was the reciprocal of the number, as long as the number was not zero. The matrix is the inverse of a square matrix A, denoted by $A^{- 1}$ , such that the product of A and $A^{- 1}$ is the identity matrix. The resulting identity matrix will be the same size as matrix A.

$A^{- 1} = \frac{1}{d e t (A)} a d j (A)$

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