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Determinants · Question 29

JEE Main 2025 (Online) 3rd April Evening ShiftNumericalHard📈 Topic in 42% papers+4
1

Let I be the identity matrix of order 3 3 and for the matrix A = bmatrix & 2 & 3 \ 4 & 5 & 6 \ 7 & -1 & 2 bmatrix, |A| = -1. Let B be the inverse of the matrix adj (A adj (A2 ) ). Then |( B + I)| is equal to

Answer

38 or -38

Solution

|A| = (10 + 6) - 2(8 - 42) + 3(-4 - 35) = 16 - 49

16 - 49 = -1 ⇒ = 3

∵ |A2| = |A|2 = 1

⇒ adj(A2) = |A2|(A2)-1 = A-2

⇒ A adj(A2) = A A-2 = A-1

⇒ adj(A-1) = |A-1| A = -A

⇒ B = (-A)-1 = -A-1

B + I = I - 3A-1 = A-1(A - 3I)

⇒ | B + I| = |A - 3I||A| = -|A - 3I|

|A - 3I| = vmatrix 0 & 2 & 3 \ 4 & 2 & 6 \ 7 & -1 & -1 vmatrix

= -2(-4 - 42) + 3(-4 - 14)

= 92 - 54 = 38

∴ | B + I| = -38 (numerical value 38)

Answer: 38 or -38

JEE Main 2025 (Online) 3rd April Evening Shift

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3

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JEE Main 2025 (Online) 29th January Morning Shift

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