Use matrix algebra properties, including 𝑨 + 𝑩 = 𝑩 + 𝑨 (commutative law for addition) 𝑨 + 0 = 𝑨 (additive identity) 𝑨 + (−𝑨) = 0 (additive inverse) 𝑨𝑰 = 𝑨 = 𝑰𝑨 (multiplicative identity) 𝑨𝑨−1 = 𝑰 = 𝑨−1𝑨 (multiplicative inverse) 𝑨(𝑩 + 𝑪) = 𝑨𝑩 + 𝑨𝑪 (left distributive law) (𝑩 + 𝑪)𝑨 = 𝑩𝑨 + 𝑪𝑨 (right distributive law)

Let \(\mathbf{P} = \begin{pmatrix} 3 & -1 \\ 2 & 4 \end{pmatrix}\), \(\mathbf{Q} = \begin{pmatrix} 5 & 2 \\ -3 & 1 \end{pmatrix}\), and \(\mathbf{R} = \begin{pmatrix} 1 & 0 \\ -2 & 3 \end{pmatrix}\). Which of the following expressions is equal to \(\mathbf{P}(\mathbf{Q} + \mathbf{R})\)?

Given matrices \(\mathbf{A} = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 5 & -2 \\ 1 & 3 \end{pmatrix}\), and \(\mathbf{C} = \begin{pmatrix} -1 & 4 \\ 2 & 0 \end{pmatrix}\), use matrix algebra properties to determine the matrix product \(\mathbf{A}(\mathbf{B} + \mathbf{C})\). Show evidence of the distributive property by calculating both \(\mathbf{A}(\mathbf{B} + \mathbf{C})\) and \(\mathbf{AB} + \mathbf{AC}\), and verify they are equal.

A small business uses matrices to track inventory flows between three warehouses. The matrix \(\mathbf{A}\) represents morning deliveries (in units) and matrix \(\mathbf{B}\) represents afternoon deliveries, where each row corresponds to a warehouse and each column to a product type. \[ \mathbf{A} = \begin{pmatrix} 12 & 8 & 5 \\ 7 & 10 & 3 \\ 9 & 6 & 11 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 4 & 9 & 2 \\ 6 & 3 & 8 \\ 5 & 7 & 4 \end{pmatrix} \] The business applies a discount matrix \(\mathbf{D}\) to adjust prices across all deliveries: \[ \mathbf{D} = \begin{pmatrix} 0.9 & 0 & 0 \\ 0 & 0.85 & 0 \\ 0 & 0 & 0.95 \end{pmatrix} \] (a) Use the left distributive law to express \(\mathbf{D}(\mathbf{A} + \mathbf{B})\) as the sum of two matrix products. (b) Calculate \(\mathbf{A} + \mathbf{B}\). (c) Use your result from part (b) to determine \(\mathbf{D}(\mathbf{A} + \mathbf{B})\). Show evidence of matrix multiplication in your solution.

Let \(\mathbf{A} = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 5 & -2 \\ 1 & 4 \end{pmatrix}\), and \(\mathbf{C} = \begin{pmatrix} -1 & 3 \\ 2 & 0 \end{pmatrix}\). a) Use matrix algebra properties to simplify \(\mathbf{A}(\mathbf{B} + \mathbf{C})\) by first expanding the expression using the distributive law. (1 mark) b) Calculate \(\mathbf{B} + \mathbf{C}\) and then determine \(\mathbf{A}(\mathbf{B} + \mathbf{C})\) by direct matrix multiplication. (1 mark) c) Use your results from parts (a) and (b) to verify that the distributive law \(\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{AB} + \mathbf{AC}\) holds for these matrices. (1 mark)