JEE Advanced 2014 · Paper 1 · Mathematics Q50

MathematicsMultiple correct+3Question 50 of 60

Let $M$ be a $2\times2$ symmetric matrix with integer entries. Then $M$ is invertible if

the first column of

M

is the transpose of the second row of

M

the second row of

M

is the transpose of the first column of

M

M

is a diagonal matrix with nonzero entries in the main diagonal

the product of entries in the main diagonal of

M

is not the square of an integer

Answer and worked solution

Official answer: C, D

Write $M=\begin{pmatrix}a&b\\b&c\end{pmatrix}$ with integer entries, $\det M=ac-b^2$ .

(A) First column $(a,b)^T$ equal to the transposed second row $(b,c)^T$ means $a=b=c$ , and then $\det M=0$ . Not invertible in general. False.
(B) The same condition. False.
(C) $b=0$ and $a,c\ne0$ give $\det M=ac\ne0$ . True.
(D) If $ac$ is not a perfect square, it cannot equal $b^2$ , so $\det M\ne0$ . True.

(C), (D)

Topics:Adjoint and inverse Special matrices

JEE Advanced 2014 · Paper 1, Mathematics question 50