Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Matrices

NDA 2018 · Mathematics · Matrices

What is the inverse of the matrix A= ( array ccc θ & θ & 0 - θ & θ...

What is the inverse of the matrix A=\left(cos⁡θsin⁡θ0−sin⁡θcos⁡θ0001\begin{array}{ccc} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right)

Options

  1. A\left(cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001\begin{array}{ccc}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right)
  2. B\left(cos⁡θ0−sin⁡θ010sin⁡θ0cos⁡θ\begin{array}{ccc}\cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta\end{array}\right)
  3. C\left(1000cos⁡θ−sin⁡θ0sin⁡θcos⁡θ\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta\end{array}\right)
  4. D\left(cos⁡θsin⁡θ0−sin⁡θcos⁡θ0001\begin{array}{ccc}\cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right)

Correct answer

A. \left(cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001\begin{array}{ccc}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right)

Step-by-step solution

A=\left[cos⁡θsin⁡θ0−sin⁡θcos⁡θ0001\begin{array}{ccc} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right] We know, AA−1=I\mathrm{AA}^{-1}=\mathrm{I}. Let us take first option (a)asA−1as A^{-1}. \therefore \mathrm{AA}^{-1}=\left[cos⁡θsin⁡θ0−sin⁡θcos⁡θ0001\begin{array}{ccc} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right]\left[cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001\begin{array}{ccc} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right]\begin{gathered}=\left[cos⁡2θ+sin⁡2θ−sin⁡θcos⁡θ+cos⁡θsin⁡θ0−sin⁡θcos⁡θ+cos⁡θsin⁡θsin⁡2θ+cos⁡2θ0001\begin{array}{ccc}\cos ^2 \theta+\sin ^2 \theta & -\sin \theta \cos \theta+\cos \theta \sin \theta & 0 \\ -\sin \theta \cos \theta+\cos \theta \sin \theta & \sin ^2 \theta+\cos ^2 \theta & 0 \\ 0 & 0 & 1\end{array}\right] \\ =\left[100010001\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]=\mathrm{I}\end{gathered}
Practice Matrices on Quantrex Academy →

More from Matrices

Consider the following statements : I. If n× n(n > 1) matrix is symmetric, then its… NDA 2026If M is a square matrix such that M^3=M , then how many values of |M| are possible ? NDA 2026If M_k= bmatrix k & k-1 k-1 & k bmatrix where k is a natural number, then what is… NDA 2026If M^ T is the transpose of a 2×2 matrix M , then which of the following is/are correct ?… NDA 2026Consider the following in respect of non-singular matrices A and B : I. (AB)^ -1 = A^ -1… NDA 2025For the following two (02) items: Let A = pmatrix θ & θ - θ & θ pmatrix . What is [adj\… NDA 2025Consider the following statements about the matrix M = pmatrix 71 & 23 & 48 57 & 28 & 29… NDA 2025If [x \ 1 \ 1] pmatrix 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 pmatrix pmatrix 1 1 x pmatrix = [45]… NDA 2025If A = pmatrix 1 & 2 & 2 2 & 1 & 2 2 & 2 & 1 pmatrix , then what is A^2 - 4A equal to? NDA 2025If A = pmatrix y & z & x z & x & y x & y & z pmatrix where x , y , z are integers, is an… NDA 2025If A = pmatrix x & y & z y & z & x z & x & y pmatrix where x , y , z are integers, is an… NDA 2025For the following two (02) items: Let A = pmatrix θ & θ - θ & θ pmatrix . What is the… NDA 2025What is the value of the determinant of the inverse of the matrix pmatrix -4 & -5 2 & 2… NDA 2025If f(θ) = pmatrix θ & θ - θ & θ pmatrix , then what is \ f(π)\ ^2 equal to? NDA 2025Let X be a matrix of order 3 × 3, Y be a matrix of order 2 × 3 and Z be a matrix of order… NDA 2024Let A and B be two square matrices of same order. If A B is a null matrix, then which one… NDA 2024 Full Matrices list Matrices JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs