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+2 votes
1.6k 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
asked in Linear Algebra by Veteran (386k points)
edited by | 1.6k views

3 Answers

+9 votes
Best answer
Square Matrix is invertible iff it is non-singular.
So both statements are same.
answered by Veteran (59.5k points)
selected 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
answered by Junior (527 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
answered by (309 points)
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