Use matrix algebra to solve matrix equations that involve matrices of up to dimension 2 × 2, including those of the form 𝑨𝑿 = 𝑩, 𝑿𝑨 = 𝑩 and 𝑨𝑿 + 𝑩𝑿 = 𝑪, with and without technology.

Solve the matrix equation for <span style="font-style:normal;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo fence="true">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="0.4em"><mtr><mtd><mn>3</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mi>X</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="0.4em"><mtr><mtd><mn>5</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd><mtd><mn>8</mn></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></math> **(a)** Find the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="0.4em"><mtr><mtd><mn>3</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable><mo fence="true">]</mo></mrow></math>. <br/> (1 mark) **(b)** Hence, solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>. <br/> (2 marks)

A manufacturing company uses matrix equations to model its production costs. Two products are manufactured, and the total production costs (in thousands of dollars) depend on two input factors tracked in matrix X. The relationships are given by: \[2\mathbf{A}\mathbf{X} + 3\mathbf{B}\mathbf{X} = \mathbf{C}\] where \[\mathbf{A} = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 1 \\ -1 & 1 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} 10 & 4 \\ 8 & 14 \end{pmatrix}\] (a) Rewrite the equation in the form \(\mathbf{M}\mathbf{X} = \mathbf{C}\), where \(\mathbf{M}\) is a single 2 × 2 matrix. State the matrix \(\mathbf{M}\). (2 marks) (b) Hence, determine the matrix \(\mathbf{X}\). You may use technology to find the inverse. (3 marks)

The table below shows the stock levels (in units) of two products at two distribution centres at the start of a financial quarter. **(a)** Express the initial stock levels as a 2 × 2 matrix S, where rows represent centres and columns represent products. (1 mark) **(b)** The transfer of stock between centres is described by the matrix equation PS = T, where P is the transfer matrix and T is the resulting stock matrix. Given P = [[0.9, 0.1], [0.15, 0.85]], calculate the matrix T. (2 marks) **(c)** The stock levels are then adjusted by a second transfer described by PT + QT = R, where Q = [[1.05, -0.05], [-0.08, 1.08]] and R is the final stock matrix. Rearrange this equation to the form (P + Q)T = R, then calculate the stock level of Product B at Centre 2 in the final stock matrix R. Give your answer to the nearest whole number. (2 marks)