Part a
Form the characteristic polynomial \(\det(A – \lambda I) = 0\) for the matrix \(A = \begin{pmatrix} 2 & p & 0 \\ 0 & 3 & 1 \\ 1 & 0 & 4 \end{pmatrix}\) in terms of \(p\) and \(\lambda\).
2×2 Matrix Addition and Subtraction
Given two 2×2 matrices, \( A = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \) and \( B = \begin{pmatrix} 5 & 7 \\ 6 & 8 \end{pmatrix} \), compute \( A+B \) and \( A-B \).
Matrix Operations in Inventory Management
A retail company uses matrices to manage their inventory levels. For part (a), perform matrix arithmetic operations to update the inventory. For part (b), interpret the resulting matrix in a real-world context.
Part a
Given matrices \(A = \begin{pmatrix} 7 & 2 \\ -1 & 3 \end{pmatrix}\) and \(B = \begin{pmatrix} 4 & 5 \\ 0 & 2 \end{pmatrix}\), calculate the matrix \(C\) defined by \(C = 2A – B\). Show all steps.