Let \(A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in R\) be written as \(P+Q\) where \(P\) is a symmetric matrix and \(Q\) is skew symmetric matrix. If \(\operatorname{det}(Q)=9\), then the modulus of the sum of all possible values of determinant of P is equal to :

JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Matrices

Let

A = [2 a 30], a \in R

be written as

P + Q

where

P

is a symmetric matrix and

Q

is skew symmetric matrix. If

det (Q) = 9

, then the modulus of the sum of all possible values of determinant of P is equal to :

Answer: (B)

For

A = [2 a 30]

the symmetric and skew symmetric parts are

P = \frac{A + A ^{T}}{2}

and

Q = \frac{A - A ^{T}}{2}

, so

P = 2 \frac{a + 3}{2} \frac{a + 3}{2} 0, Q = 0 \frac{a - 3}{2} \frac{3 - a}{2} 0

Then

det Q = - \frac{( 3 - a ) ( a - 3 )}{4} = \frac{( a - 3 ) ^{2}}{4} = 9

gives

a = 9

a = - 3

. Since

det P = - \frac{( a + 3 ) ^{2}}{4}

, these two values give

det P = - 36

and

det P = 0

, whose sum is

- 36

. The modulus of that sum is

36

JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Single correct option