25 votes 25 votes Let $X$ be a square matrix. Consider the following two statements on $X$. $X$ is invertible Determinant of $X$ is non-zero Which one of the following is TRUE? I implies II; II does not imply I II implies I; I does not imply II I does not imply II; II does not imply I I and II are equivalent statements Linear Algebra gatecse-2019 engineering-mathematics linear-algebra determinant 1-mark + – Arjun asked Feb 7, 2019 retagged Nov 30, 2022 by Lakshman Bhaiya Arjun 10.4k views answer comment Share Follow See all 2 Comments See all 2 2 Comments reply val_pro20 commented Nov 4, 2019 reply Follow Share What if matrix is not square matrix 0 votes 0 votes Lakshman Bhaiya commented Nov 29, 2019 reply Follow Share @val_pro20 The inverse of the non-square matrix doesn't exist. 5 votes 5 votes Please log in or register to add a comment.
Best answer 27 votes 27 votes Square Matrix is invertible iff it is non-singular. So both statements are same.Answer is (D). Digvijay Pandey answered Feb 7, 2019 edited May 13, 2019 by Krithiga2101 Digvijay Pandey comment Share Follow See all 4 Comments See all 4 4 Comments reply Lakshman Bhaiya commented Feb 11, 2019 i edited by Lakshman Bhaiya Feb 6, 2020 reply Follow Share $X^{-1}=\dfrac{Adj(X)}{\mid X\mid}$ $I)$ 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/ 10 votes 10 votes Raghav Khajuria commented Feb 11, 2019 reply Follow Share If the option I implies II and II implies I given then I think it would be much suitable 0 votes 0 votes Lakshman Bhaiya commented Feb 11, 2019 reply Follow Share 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. $P\Longleftrightarrow Q\equiv (P\Longrightarrow Q)\wedge (Q\Longrightarrow P)$ 5 votes 5 votes Raghav Khajuria commented Feb 11, 2019 reply Follow Share Thanks 0 votes 0 votes Please log in or register to add a comment.
6 votes 6 votes if a square matrix is invertible then it's determinant is non zero and vice versa so, (D) is correct option Dharmendra Tiwari answered Feb 7, 2019 Dharmendra Tiwari comment Share Follow See all 0 reply Please log in or register to add a comment.
3 votes 3 votes 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 Sukhbir Singh answered Feb 7, 2019 Sukhbir Singh comment Share Follow See all 0 reply Please log in or register to add a comment.
2 votes 2 votes 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 gaurav1.yuva answered Apr 23, 2019 gaurav1.yuva comment Share Follow See all 0 reply Please log in or register to add a comment.