The Gateway to Computer Science Excellence
First time here? Checkout the FAQ!
x
+3 votes
1.8k views

Let $X$ be a square matrix. Consider the following two statements on $X$.

  1. $X$ is invertible
  2. Determinant of $X$ is non-zero

Which one of the following is TRUE?

  1. I implies II; II does not imply I
  2. II implies I; I does not imply II
  3. I does not imply II; II does not imply I
  4. I and II are equivalent statements
in Linear Algebra by Veteran (416k points)
edited by | 1.8k views

4 Answers

+9 votes
Best answer
Square Matrix is invertible iff it is non-singular.
So both statements are same.Answer is (D).
by Veteran (60k points)
edited by
0

$X^{-1}=\frac{Adj(X)}{|X|}$

$I)$ If $X^{-1}$ exist then $|X|\neq0$.

        $X^{-1}\implies|X|\neq0$

$X^{-1}$ $|X|\neq0$.  $X^{-1}\implies|X|\neq0$
T T T
T F F
F T T
F F T


$II)$ If $|X|\neq 0$ then $X^{-1}$ exist.

            $|X|\neq 0\implies X^{-1}$

$|X|\neq 0$ $X^{-1}$ $|X|\neq 0\implies X^{-1}$
T T T
T F F
F T T
F F T

we can write like this  $X^{-1}$ exist if only if (or) iff $|X|\neq0$

$X^{-1}$ $|X|\neq0$ $|X|\neq 0\Longleftrightarrow X^{-1}$
T T T
T F F
F T F
F F T

$I)$ and $II)$ Both are equivalent

0
If the option I implies II and II implies I given then I think it would be much suitable
0
Yes, you think in a simple way, but IIT professor think some other way

actually what you write is correct but what they ask is also correct.

$P\Longleftrightarrow Q\equiv (P\Longrightarrow Q)\wedge (Q\Longrightarrow P)$
0
Thanks
+5 votes
if a square matrix is invertible then it's determinant is non zero

and vice versa

so, (D) is correct option
by Junior (511 points)
+1 vote
Option D is right. As Inverse(A) = Adj(A) / Mod(A)

 

Therefore, if Mod(A) is 0, the Inverse of a matrix cannot be calculated. Therefore both statements are equivalent to each other
by (341 points)
0 votes

The inverse of a square matrix A, sometimes called a reciprocal matrix, is a matrix A^(-1) such that

AA^(-1)=I,

 

where I is the identity matrix. Courant and Hilbert (1989, p. 10) use the notation A^_ to denote the inverse matrix.

square matrix A has an inverse iff the determinant |A|!=0 (Lipschutz 1991, p. 45). The so-called invertible matrix theorem is major result in linear algebra which associates the existence of a matrix inverse with a number of other equivalent properties. 

 

http://mathworld.wolfram.com/MatrixInverse.html

by (437 points)
Answer:

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
49,845 questions
54,787 answers
189,449 comments
80,540 users