Question two: The following Matrix gives the number of units of three product ( \( \mathrm{M}, \mathrm{N} \) and O ), that can be processed per hour on three machines ( \( \mathrm{I}, \mathrm{II} \) and III) \[ \begin{array}{lll}\text { I } & \text { II } & \text { III }\end{array} \] \[ N\left[\begin{array}{lll}10 & 12 & 15 \\ 20 & 11 & 20 \\ 16 & 18 & 14\end{array}\right] \] Determine by using matrix algebra, how many units of each product can be produced, if the hours available on machine I, II and III are 50,40 and 42

To find the number of units of products \( \mathrm{M}, \mathrm{N}, \) and \( \mathrm{O} \) that can be produced using matrix algebra, we can set up a system of equations based on the given matrix and the available hours for each machine. Let \( x_1, x_2, \) and \( x_3 \) be the number of units produced of products \( \mathrm{M}, \mathrm{N}, \) and \( \mathrm{O} \), respectively. We will use the matrix of processing capabilities and the hours available for each machine to create the equation: \[ \begin{bmatrix} 10 & 12 & 15 \\ 20 & 11 & 20 \\ 16 & 18 & 14 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 50 \\ 40 \\ 42 \end{bmatrix} \] By solving this matrix equation using techniques such as Gaussian elimination or using an inverse matrix, you'll find the values of \( x_1, x_2, \) and \( x_3 \). One fun historical tip: the concept of using linear algebra and matrix methods to solve real-world problems dates back to ancient civilizations! While matrices became formalized in the 19th century, similar systems can be traced back to various historical applications like resource allocation in agriculture or inventory management in trade. For practical application in today's world, businesses regularly utilize matrix algebra for resource optimization. For instance, manufacturers might use these calculations to determine how many products to produce based on limited machine time and different product line efficiencies. This not only saves time but also maximizes profit, helping to streamline operations in a fun, dynamic way!