JEE Main · 2010 · 11 Apr 2010 · Offline · Matrices

The number of

3 \times 3

non-singular matrices, with four entries as 1 and all other entries as 0, is

(A)5
(B)6
(C)at least 7✓ Correct
(D)less than 4

Answer: (C)

Solution

We want

3 \times 3

matrices with exactly four entries equal to

1

and the rest

0

that are non-singular. One ready family keeps the three diagonal

1

s, which alone give the identity, and adds a single

1

in any one of the six off-diagonal places; each such matrix is triangular in effect with non-zero determinant, already giving six non-singular matrices. Further non-singular arrangements exist with the four

1

s placed differently, so the total exceeds six. The number of such matrices is at least seven.

JEE Main · 2010 · 11 Apr 2010 · Offline · Single correct option

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