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JEE Main · 2025 · 04 Apr 2025 · Shift 2 · Matrices

Let the matrix

A = 110001010

satisfy

A^{n} = A^{n - 2} + A^{2} - I

for

n \geq 3

. Then the sum of all the elements of

A^{50}

is :-

(A)53✓ Correct
(B)52
(C)39
(D)44

Answer: (A)

Solution

Applying

A^{n} = A^{n - 2} + (A^{2} - I)

repeatedly from

n = 50

down to

n = 2

adds the block

A^{2} - I

twenty-four times, so

A^{50} = 25 A^{2} - 24 I

. With

A^{2} = 111010001

, this gives

A^{50} = 12525010001

, whose entries sum to

53

.

JEE Main · 2025 · 04 Apr 2025 · Shift 2 · Single correct option

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