JEE Advanced · 2020 · 27 Sep 2020 · Paper 2 · Matrices

The trace of a square matrix is defined to be the sum of its diagonal entries. If

A

is a 2 × 2 matrix such that the trace of

A

is 3 and the trace of

A^{3}

is -18, then the value of the determinant of

A

is ____

Answer: 5

Solution

Let

A = [a c b d]

. Given

trace (A) = a + d = 3

and

trace (A^{3}) = - 18

. Computing

A^{3}

and taking its trace:

a^{3} + d^{3} + 3 ab c + 3 b c d = - 18

. This simplifies to

a^{3} + d^{3} + 3 b c (a + d) = - 18

, so

a^{3} + d^{3} + 9 b c = - 18

. Using the identity

a^{3} + d^{3} = (a + d) ((a + d)^{2} - 3 a d) = 3 (9 - 3 a d) = 27 - 9 a d

, we get

27 - 9 a d + 9 b c = - 18

, which gives

a d - b c = 5

. Therefore,

det (A) = 5

JEE Advanced · 2020 · 27 Sep 2020 · Paper 2 · Integer answer