IIT-JEE 2009 · Paper 1 · Mathematics Q35

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Paragraph for Question Nos. 33 to 35

Let $\mathcal A$ be the set of all $3\times3$ 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 $\mathcal A$ for which the system of linear equations
$A\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}$
is inconsistent, is

more than 2

Answer and worked solution

Official answer: B

Singular matrices (6 of them) are consistent or not depending on the right side $(1,0,0)^T$ .

$d=3$ , pair {12}: rows give $x+y=1$ and $x+y=0$ : inconsistent. Pair {13}: $x+z=1$ and $x+z=0$ : inconsistent. Pair {23}: $x=1$ , $y+z=0$ twice: consistent.
$d=1$ with pairs to the diagonal index: $a_{11}=1$ : $x+y+z=1$ , $x=0$ , $x=0$ : consistent. $a_{22}=1$ : $y=1$ and $y=0$ : inconsistent. $a_{33}=1$ : $z=1$ and $z=0$ : inconsistent.

Inconsistent: 4, more than 2.

(B)

Topics:Systems of linear equations

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IIT-JEE 2009 · Paper 1, Mathematics question 35