Let $\mathscr{A}$ be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0. The number of matrices $A$ in $\mathscr{A}$ for which the system of linear equations $$ A\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right] $$ has a unique solution, is

(C) at least 7 but less than 10

JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Matrices

Let

A

be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

The number of matrices

A

A

for which the system of linear equations

A x y z = 100

has a unique solution, is

(A)less than 4
(B)at least 4 but less than 7✓ Correct
(C)at least 7 but less than 10
(D)at least 10

Answer: (B)

Solution

For a unique solution, we need

det (A) \neq = 0

. From the 12 matrices in

A

, we need to count those with non-zero determinant.

The solution in the source identifies 6 matrices with

det (A) \neq = 0

by systematically checking different patterns:

Matrices with specific zero patterns that maintain non-singularity
The key is that certain positions of zeros preserve linear independence of rows

The count is 6, which falls in the range "at least 4 but less than 7".

JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Single correct option

Practise more: all JEE Advanced PYQs on Matrices →

JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Matrices

Solution

More JEE Advanced questions on Matrices