1. **Problem Statement:** Calculate the characteristic equation of matrix a): $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & -2 & 6 \\ 0 & 0 & -3 \end{bmatrix}$$ 2. **Formula:** The characteristic equation of a matrix $A$ is given by: $$\det(A - \lambda I) = 0$$ where $\lambda$ is an eigenvalue and $I$ is the identity matrix. 3. **Step-by-step calculation for matrix a):** - Write $A - \lambda I$: $$\begin{bmatrix} 1-\lambda & 2 & 3 \\ 0 & -2-\lambda & 6 \\ 0 & 0 & -3-\lambda \end{bmatrix}$$ - Since this is an upper triangular matrix, the determinant is the product of diagonal elements: $$\det(A - \lambda I) = (1-\lambda)(-2-\lambda)(-3-\lambda)$$ - Set the determinant equal to zero for the characteristic equation: $$ (1-\lambda)(-2-\lambda)(-3-\lambda) = 0 $$ 4. **Final characteristic equation for matrix a):** $$ (1-\lambda)(-2-\lambda)(-3-\lambda) = 0 $$ This can be expanded if needed, but the factored form clearly shows eigenvalues. --- Since the user asked for both a) and b) but per instructions only the first problem is solved completely, the count of distinct problems is 2.