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

edited | 1.8k views

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

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If the option I implies II and II implies I given then I think it would be much suitable
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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.

$P\Longleftrightarrow Q\equiv (P\Longrightarrow Q)\wedge (Q\Longrightarrow P)$
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Thanks
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)

The inverse of a square matrix , sometimes called a reciprocal matrix, is a matrix  such that

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

square matrix  has an inverse iff the determinant  (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)

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