1. **Problem Statement:** We are given a 3x3 scale matrix $A$ with distinct non-zero diagonal elements $a_{11}$, $a_{22}$, and $a_{33}$. We need to calculate the determinant of this matrix. 2. **Definition of a Scale Matrix:** A scale matrix is a diagonal matrix where all off-diagonal elements are zero and diagonal elements represent scaling factors. For matrix $A$: $$ A = \begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{bmatrix} $$ 3. **Formula for Determinant of a Diagonal Matrix:** The determinant of a diagonal matrix is the product of its diagonal elements: $$\det(A) = a_{11} \times a_{22} \times a_{33}$$ 4. **Step-by-step Computation:** - Write the matrix: $$ A = \begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{bmatrix} $$ - Calculate determinant: $$\det(A) = a_{11} \times a_{22} \times a_{33}$$ 5. **Explanation:** Since all off-diagonal elements are zero, the determinant simplifies to the product of the diagonal elements. This is a key property of diagonal matrices. **Final answer:** $$\det(A) = a_{11} a_{22} a_{33}$$