GATE CSE 2019 | Question: 9

Question

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?

• A. I implies II; II does not imply I
• B. II implies I; I does not imply II
• C. I does not imply II; II does not imply I
• D. I and II are equivalent statements

Comments

What if matrix is not square matrix

---

@val_pro20

The inverse of the non-square matrix doesn't exist.

Answer 1

Square Matrix is invertible iff it is non-singular.
So both statements are same.Answer is (D).

Comments

$X^{-1}=\dfrac{Adj(X)}{\mid X\mid}$

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

        $X^{-1}\implies \mid X\mid \:\neq0$

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

            $\mid X\mid\:\neq 0\implies X^{-1}$

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

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

 

$X^{-1}$	$\mid X \mid \neq0$	$\big(X^{-1}\implies \mid X \mid \neq0\big)\wedge \big(\mid X \mid \neq0\implies X^{-1}\big) $ $\equiv\:\: \mid X\mid \:\:\neq 0\Longleftrightarrow X^{-1}$
$T$	$T$	$T$
$T$	$F$	$F$
$F$	$T$	$F$
$F$	$F$	$T$

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

Reference: https://brilliant.org/wiki/matrices/

---

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

---

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.

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

---

Thanks

Answer 2

if a square matrix is invertible then it's determinant is non zero

and vice versa

so, (D) is correct option

Answer 3

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

Answer 4

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.

A 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