1. The problem is to understand the properties of the given 3x3 matrix: $$\begin{bmatrix} 1 + a^2 - b^2 & 2ab & -2b \\ 2ab & 1 - a^2 + b^2 & 2a \\ 2b & -2a & 1 - a^2 - b^2 \end{bmatrix}$$ 2. This matrix has entries that are quadratic polynomials in variables $a$ and $b$, with mixed terms like $2ab$ and linear terms like $-2b$, $2a$. 3. Such a matrix form indicates it might be related to orthogonal transformations or rotations in three dimensions since the sums of squares appear prominently. 4. Let's check if it is an orthogonal matrix by verifying if the matrix multiplied by its transpose equals the identity matrix. This involves calculating: $$M M^{T} = I$$ 5. We note that the expressions for each element suggest the sums of squares $a^2 + b^2$ play a role. 6. Without loss of generality, suppose $a^2 + b^2 = r^2$. Substituting this into elements indicates the diagonal entries involve $1 \pm r^2$ and mixed terms involving products of $a$ and $b$. 7. This matrix is actually a parameterization of a rotation matrix in 3D space depending on two variables $a$ and $b$, reminiscent of Rodrigues' rotation formula or quaternion representations. 8. In summary, the matrix is a 3D rotation matrix parameterized by $a$ and $b$. It is orthogonal and preserves length, i.e., $M M^{T} = I$. Final conclusion: The matrix is an orthogonal matrix dependent on parameters $a$ and $b$, representing a rotation in 3D space.