157 Use the method \( \left[A \mid I_{n}\right] \rightarrow\left[I_{n} \mid A^{-1}\right] \) to find the inverses, where possible, of the \( m \) below. \[ \begin{array}{llll}\text { (h) } A=\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 1 \\ 1 & 0 & 1\end{array}\right]\end{array} \]

To find the inverse of the matrix \( A \) using the method \( \left[A \mid I_{n}\right] \rightarrow\left[I_{n} \mid A^{-1}\right] \), we will augment the matrix \( A \) with the identity matrix \( I_3 \) and perform row operations to transform \( A \) into \( I_3 \). The resulting augmented matrix will then give us \( A^{-1} \). The matrix \( A \) is given by: \[ A = \begin{bmatrix} \cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 1 \\ 1 & 0 & 1 \end{bmatrix} \] We augment this matrix with the identity matrix: \[ \left[A \mid I_3\right] = \left[\begin{array}{ccc|ccc} \cos \theta & \sin \theta & 0 & 1 & 0 & 0 \\ \sin \theta & \cos \theta & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 \end{array}\right] \] Now, we will perform row operations to convert the left side into the identity matrix. ### Step 1: Make the first column a pivot column We can start by making the first row the pivot row. We can swap rows if necessary, but in this case, we can use the third row to eliminate the first column entries in the first and second rows. 1. **Row 1**: \( R_1 \leftarrow R_1 \) (no change) 2. **Row 2**: \( R_2 \leftarrow R_2 - \sin \theta R_1 \) 3. **Row 3**: \( R_3 \leftarrow R_3 - \cos \theta R_1 \) Calculating these operations: - For \( R_2 \): \[ R_2 = \begin{bmatrix} \sin \theta & \cos \theta & 1 & 0 & 1 & 0 \end{bmatrix} - \sin \theta \begin{bmatrix} \cos \theta & \sin \theta & 0 & 1 & 0 & 0 \end{bmatrix} \] \[ = \begin{bmatrix} 0 & \cos \theta - \sin^2 \theta & 1 & -\sin \theta & 1 & 0 \end{bmatrix} \] - For \( R_3 \): \[ R_3 = \begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 1 \end{bmatrix} - \cos \theta \begin{bmatrix} \cos \theta & \sin \theta & 0 & 1 & 0 & 0 \end{bmatrix} \] \[ = \begin{bmatrix} 0 & -\cos \theta \sin \theta & 1 & -\cos \theta & 0 & 1 \end{bmatrix} \] Now our augmented matrix looks like this: \[ \left[\begin{array}{ccc|ccc} \cos \theta & \sin \theta & 0 & 1 & 0 & 0 \\ 0 & \cos \theta - \sin^2 \theta & 1 & -\sin \theta & 1 & 0 \\ 0 & -\cos \theta \sin \theta & 1 & -\cos \theta & 0 & 1 \end{array}\right] \] ### Step 2: Normalize the second row Next, we need to make the second row's leading coefficient equal to 1. We can do this by dividing the second row by \( \cos \theta - \sin^2 \theta \) (assuming it is not zero). ### Step 3: Eliminate the second column entries We will then eliminate the second column entries in the first and third rows using the second row. ### Step 4: Normalize the third row Finally, we will normalize the third row and eliminate any remaining entries above it. ### Final Result After performing all necessary row operations, we will arrive at the identity matrix on the left side, and the right side will give us \( A^{-1} \). The final form of \( A^{-1} \) will depend on the specific values of \( \theta \). However, the general approach remains the same, and you can compute the exact entries of \( A^{-1} \) based on the row operations performed. If