JEE Main · 2008 · 27 Apr 2008 · Offline · Matrices

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

(A)If $det A = \pm 1$ , then $A^{- 1}$ exists but all its entries are not necessarily integers
(B)If $det A \neq = \pm 1$ , then $A^{- 1}$ exists and all its entries are non-integers
(C)If $det A = \pm 1$ , then $A^{- 1}$ exists and all its entries are integers✓ Correct
(D)If $detA = \pm 1$ , then $A^{- 1}$ need not exist

Answer: (C)

Solution

If every entry of

A

is an integer, then every cofactor, and hence every entry of the adjugate, is an integer. The inverse is

A^{- 1} = \frac{1}{det A} adj (A)

, so when

det A = \pm 1

the division introduces no fractions and

A^{- 1}

exists with all entries integers.

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option