Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202623 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
23 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Matrices

JEE Main 2026 · 23 January · Evening Shift · Mathematics · Matrices

Let A=[ array ccc 0 & 2 & -3 -2 & 0 & 1 3 & -1 & 0 array ] and B be...

Let A=\left[02−3−2013−10\begin{array}{ccc}0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0\end{array}\right] andB be a matrix such that B(I−A)=I+AB(I-A)=I+A. Then the sum of the diagonal elements of BTB\mathrm{B}^{\mathrm{T}} \mathrm{B} is equal to ____\_\_\_\_.

Correct answer

0

Step-by-step solution

AA is skew-symmetric (AT=−AA^T = -A).
From B(I−A)=I+AB(I-A) = I+A:
B=(I+A)(I−A)−1B = (I+A)(I-A)^{-1}.
BT=((I−A)T)−1(I+A)T=(I+A)−1(I−A)B^T = ((I-A)^T)^{-1}(I+A)^T = (I+A)^{-1}(I-A).
Since AA is skew-symmetric, (I−A)(I-A) and (I+A)(I+A) commute:
(I−A)(I+A)=I−A2=(I+A)(I−A)(I-A)(I+A) = I - A^2 = (I+A)(I-A).
BTB=(I+A)−1(I−A)(I+A)(I−A)−1=(I+A)−1(I+A)(I−A)(I−A)−1=IB^TB = (I+A)^{-1}(I-A)(I+A)(I-A)^{-1} = (I+A)^{-1}(I+A)(I-A)(I-A)^{-1} = I.
Trace of BTB=tr(I)=3B^TB = \text{tr}(I) = 3.
Practice Matrices on Quantrex Academy →

More from Matrices

Which one of the following matrices can be obtained by performing elementary row… JEE Advanced 2026Consider the matrix M = bmatrix 2 & -1 1 & 0 bmatrix . Let p, q, r, s, a, b, c and d be… JEE Advanced 2026For real numbers α, β, γ, δ and μ , consider the matrix M = bmatrix α & 1 2 & - 1 2 1 3 &… JEE Advanced 2026Let ℝ denote the set of all real numbers and let i = -1 . Consider the matrices S =… JEE Advanced 2026Let A = bmatrix α & 1 & 2 2 & 3 & 0 0 & 4 & 5 bmatrix and B = bmatrix 1 & 0 & 0 0 & -5α &… JEE Main 2026Let A = bmatrix 1 & 0 & 0 3 & 1 & 0 9 & 3 & 1 bmatrix and B = [b_ ij ] , 1 ≤ i, j ≤ 3 .… JEE Main 2026Let A = bmatrix -1 & 1 & -1 1 & 0 & 1 0 & 0 & 1 bmatrix satisfy A^2 + α(adj(adj(A))) +… JEE Main 2026Let M be a 3 × 3 matrix such that M pmatrix 1 0 0 pmatrix = pmatrix 1 2 3 pmatrix , M… JEE Main 2026Let A be a 3 × 3 matrix such that A^T bmatrix 1 0 1 bmatrix = bmatrix 5 2 2 bmatrix , A^T… JEE Main 2026Let A= bmatrix 1 & 2 & 7 4 & -2 & 8 3 & 8 & -7 bmatrix and (A-α I)=0 , where α is a real… JEE Main 2026Let S = \ A = bmatrix a & b c & d bmatrix : a, b, c, d ∈ \ 0, 1, 2, 3, 4\ and A^2 - 4A +… JEE Main 2026Let A = bmatrix 1 & 1 & 2 -2 & 0 & 1 1 & 3 & 5 bmatrix . Then the sum of all elements of… JEE Main 2026Consider the matrices A = bmatrix 2 & -2 4 & -2 bmatrix and B = bmatrix 3 & 9 1 & 3… JEE Main 2026Let A = bmatrix 1 & 2 1 & α bmatrix and B = bmatrix 3 & 3 β & 2 bmatrix . If A^2 - 4A + I… JEE Main 2026Let A=[ array ll 3 & -4 1 & -1 array ] and B be two matrices such that A^ 100 =100 B+I .… JEE Main 2026Let A, B and C be three 2 × 2 matrices with real entries such that B=(I+A)^ -1 and A+C=I… JEE Main 2026 Full Matrices list Matrices JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs