Concept:Expansion of determinant \(\begin{vmatrix}a_{11} & b_{12} & c_{13} \\ a_{21} & b_{22} & c_{23} \\ a_{31} & b_{32} & c_{33} \end{vmatrix}\) is given by ? = a11 a22 a33 - a11 a23 a32 - a12 a21 a33 + a12 a23 a31 + a13 a21 a32 - a13 a31 a22A(adj A) = ? A ? I, where I is the identity matrix.Calculation:Given: \(A = \begin{bmatrix} 2 \sin ? & \cos ? & 0 \\ -2\cos ? & \sin ? & 0 \\ -1 & 1 & 1 \end{bmatrix} \) and I is the identity matrix.? \(\mid A\mid = \begin{vmatrix} 2 \sin ? & \cos ? & 0 \\ -2\cos ? & \sin ? & 0 \\ -1 & 1 & 1 \end{vmatrix}\)Expanding about a3,3, we get,? ? A ? = 1{(2 sin? × sin?) + 2 cos2?}? ? A ? = 2(2 sin2? + 2 cos2?)? ? A ? = 2(sin2? + cos2?)? ? A ? = 2 [sin2? + cos2? = 1]? A(adj A) = ? A ? IPutting the value of ? A ? = 2,? A(adj A) = 2I, where I is the identity matrix.? A(adj A) = 2I