Define and use addition and subtraction of matrices, scalar multiplication, matrix multiplication, multiplicative identity and multiplicative inverse.

A manufacturing company uses matrix multiplication to track production costs across two factories. The matrix \(\mathbf{P} = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}\) represents the number of hours required by Factory A and Factory B (rows) to produce two products X and Y (columns). The matrix \(\mathbf{C} = \begin{pmatrix} 45 \\ 30 \end{pmatrix}\) represents the hourly cost (in dollars) for products X and Y respectively. a) Determine the matrix product \(\mathbf{PC}\) and explain what this product represents in the context of the manufacturing scenario. (2 marks) b) Use your result from part (a) to identify which factory has the higher total production cost. (1 mark)

Determine the single matrix \(\mathbf{C}\) that transforms \(\mathbf{u}\) directly to \(\mathbf{w}\).

Given \(\mathbf{A} = \begin{pmatrix} 3 & -1 \\ 2 & 4 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 5 & 2 \\ -3 & 1 \end{pmatrix}\), determine the matrix \(\mathbf{C}\) that satisfies the equation \(2\mathbf{A} - \mathbf{C} = \mathbf{B}\). Show evidence of the matrix operations used in your solution.

Consider the matrix \(\mathbf{A} = \begin{pmatrix} 3 & 2 \\ 5 & 4 \end{pmatrix}\). a) Use the results from your calculation of the determinant to determine the multiplicative inverse \(\mathbf{A}^{-1}\) of matrix \(\mathbf{A}\). (2 marks) b) Verify that your result from part (a) is correct by showing that \(\mathbf{A} \mathbf{A}^{-1}\) equals the multiplicative identity matrix. (1 mark)