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NDA 1 2024 Maths, question 29

Topic: Matrices and determinants

Q29

If A=[sin⁡2θ2sin⁡2θ−10cos⁡2θ2sin⁡θcos⁡θ0001]A=\begin{bmatrix} \sin 2\theta & 2\sin ^{2}\theta-1 & 0 \\ \cos 2\theta & 2\sin \theta\cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}, then which of the following statements is/are correct ?

  1. A−1=adjAA^{-1}=\text{adj}A

  2. AA is skew-symmetric matrix

  3. A−1=ATA^{-1}=A^{T}

Select the correct answer using the code given below :

  1. (a) 1 only
  2. (b) 1 and 2
  3. (c) 1 and 3Correct answer
  4. (d) 2 and 3

Solution.

Answer: (c) 1 and 3

  1. Write s=sin2θ and c=cos2θ.
  2. The upper block is [[s,−c],[c,s]], which has determinant s²+c²=1 and orthonormal columns.
  3. With the final diagonal entry 1, det A=1 and A⁻¹=Aᵀ=adj A.
  4. It is not skew-symmetric because its last diagonal entry is 1.
  5. Thus 1 and 3 hold.