Matrix Sum in Graph Analysis
Given two directed graphs with adjacency matrices \( A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \) and \( B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \), compute the matrix sum \( C = A + B \). Explain how the resulting matrix represents the connectivity of the […]
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 \( A-B \).
Solving a Matrix Equation with Parameters
Given matrices \(A = \begin{pmatrix} 2 & 3 \\ 0 & k \end{pmatrix}\) and \(B = \begin{pmatrix} 5 & 7 \\ 1 & 4 \end{pmatrix}\), form the matrix expression \(A + cB\). Determine the unknown scalar \(c\) and parameter \(k\) such that the (2,1) entry of \(A + cB\) is 1 and the (1,2) entry […]
Classifying Matrix Shapes and Dimensions
A matrix is a rectangular array of numbers. Consider the matrix \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix} \). Answer the following:
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.
Budget Matrix Operations
A company uses a 2×2 budget matrix \(A = \begin{pmatrix} 5000 & 7000 \\ 3000 & 4000 \end{pmatrix}\) to represent its departmental allocations. Due to an anticipated revenue increase, the company first increases all allocations by 10% and then adds a bonus allocation given by matrix \(B = \begin{pmatrix} 200 & 300 \\ 0 & […]
Evaluating Combined Sales with Matrix Arithmetic
A restaurant tracks the daily sales of two menu items from two branches over two consecutive days. The sales are recorded in matrices. For Day 1, the sales are given by: \( A = \begin{pmatrix} 10 & 15 \\ 20 & 25 \end{pmatrix} \) and for Day 2 by: \( B = \begin{pmatrix} 8 & […]