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25 votes
25 votes

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
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7 Answers

Best answer
27 votes
27 votes
Square Matrix is invertible iff it is non-singular.
So both statements are same.Answer is (D).
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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
2 votes
2 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

Answer:

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