Matrices

JEE Main 2025 (Online) 24th January Morning ShiftNumericalMedium📈 Topic in 5% papers+4

Ques. 1JEE Main 2025, 24 January (Morning Shift)

Let A be a 3 3 matrix such that XTAX=O for all nonzero 3 1 matrices X= bmatrixx\ \ bmatrix. If A bmatrix1\\1\\1 bmatrix= bmatrix1\\4\\-5 bmatrix, A bmatrix1\\2\\1 bmatrix= bmatrix0\\4\\-8 bmatrix, and (adj(2(A+I)))=2 3 5 , , , N, then 2+ 2+ 2 is

Answer

Solution

XTAX=0 for all X A+AT=O: A= bmatrix0&a&b\\-a&0&c\\-b&-c&0 bmatrix

Given products: a+b=1, 2a+b=0, -a+c=4 a=-1, b=2, c=3

|A+I|=1+a2+b2+c2=15 |2(A+I)|=23 15=120

|adj M|=|M|2=1202=26 32 52

2+ 2+ 2=36+4+4=44

Answer: 44

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Ques. 2JEE Main 2025, 24 January (Evening Shift)

Let A=[aij] be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P(E) is :

Ques. 3JEE Main 2025, 23 January (Evening Shift)

Let A=[aij] be a 3 3 matrix such that A bmatrix0\\1\\0 bmatrix= bmatrix0\\0\\1 bmatrix, A bmatrix4\\1\\3 bmatrix= bmatrix0\\1\\0 bmatrix and A bmatrix2\\1\\2 bmatrix= bmatrix1\\0\\0 bmatrix, then a23 equals :

2
-1
1
0

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