Let α and β be real numbers. Consider a 3 × 3 matrix A such that A^2 = 3A + α I. If A^4 = 21A + β I, then

Let α and β be real numbers. Consider a 3 × 3 matrix A such… — JEE Main 2023 Mathematics PYQ

JEE Main 2023 · 29 Jan · Shift 1

Let

\alpha

and

\beta

be real numbers. Consider a

3 \times 3

matrix

A

such that

A^2 = 3A + \alpha I

. If

A^4 = 21A + \beta I

, then

A.
$\alpha = 1$
B.
$\beta = -8$
C.
$\beta = 8$
D.
$\alpha = 4$

Show answer and solution

Correct option: B

Solution

From A² = 3A + αI, compute A⁴ = (A²)² = (3A + αI)² = 9A² + 6αA + α²I.

Substitute A² = 3A + αI: A⁴ = 9(3A + αI) + 6αA + α²I = (27 + 6α)A + (9α + α²)I.

This must equal 21A + βI, so:
27 + 6α = 21 → α = -1
β = 9α + α² = 9(-1) + 1 = -8.

Thus β = -8.

[JEE Main 2023 · 29 Jan · Shift 1]

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