Word Count : 631

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To determine the values of \(\lambda\) for which the given matrix is orthogonal, we need to ensure that the matrix \(A\) satisfies the condition \(A^T A = I\), where \(A^T\) is the transpose of \(A\) and \(I\) is the identity matrix. The given matrix is: \[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda^2 \end{bmatrix} \] ### Step 1: Compute \(A^T A\) First, compute the transpose of \(A\), denoted \(A^T\): \[ A^T = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda^2 \end{bmatrix} \] Now, multiply \(A^T\) by \(A\): \[ A^T A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda^2 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda^2 \end{bmatrix} = \begin{bmatrix} 1 \cdot 1 ____ __________ ___ ___ ______ ________ ___.
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