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JEE Main · 2026 · 21 Jan 2026 · Shift 1 · Matrices

For some

α, β \in R

, let

A = [α 1 22]

and

B = [11 1 β]

be such that

A^{2} - 4 A + 2 I = B^{2} - 3 B + I = O

. Then

(det (adj (A^{3} - B^{3})))^{2}

is equal to

Answer: 225

Solution

The relations

A^{2} - 4 A + 2 I = O

and

B^{2} - 3 B + I = O

are the Cayley–Hamilton equations, fixing the traces and determinants:

α = 2, β = 2

, so

A = (2122)

and

B = (1112)

. Reducing powers,

A^{3} = 14 A - 8 I = (20142820)

and

B^{3} = 8 B - 3 I = (58813)

, so

A^{3} - B^{3} = (156207)

with determinant

- 15

. For a

2 \times 2

matrix

det (adj M) = det M

, hence

(det (adj (A^{3} - B^{3})))^{2} = (- 15)^{2} = 225

.

JEE Main · 2026 · 21 Jan 2026 · Shift 1 · Integer answer

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