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JEE Main · 2016 · 03 Apr 2016 · Offline · Matrices

If

A = [5 a 3 - b 2]

and A adj

A = A A^{T}

, then

5 a + b

is equal to:

(A)5✓ Correct
(B)4
(C)13
(D)-1

Answer: (A)

Solution

Since

A adj (A) = ∣ A ∣ I

, the product

A A^{T}

must also be a scalar matrix. Here

A A^{T} = [25 a^{2} + b^{2} 15 a - 2 b 15 a - 2 b 13]

and

∣ A ∣ I = (10 a + 3 b) I

, so the off-diagonal entries force

15 a - 2 b = 0

and the diagonal gives

10 a + 3 b = 13

. From the first,

b = \frac{15 a}{2}

; substituting,

10 a + \frac{45 a}{2} = 13

gives

a = \frac{2}{5}

and

b = 3

. Therefore

5 a + b = 5 \cdot \frac{2}{5} + 3 = 5

.

JEE Main · 2016 · 03 Apr 2016 · Offline · Single correct option

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