Why It Matters: Solve Systems with Matrices
Why solve systems with matrices?
In a previous module, Polynomial and Rational Expressions, we explored a scenario in which a furniture company produces chairs and tables. Now suppose that our company has two workshops that both produce the tables and chairs, one in Atlanta, and the other in Boston. Furthermore, the two workshops make the furniture at different rates. The table below shows the maximum number of chairs and tables that can be made each day at the two workshops.Atlanta | Boston | |
Chairs | 20 per day | 30 per day |
Tables | 15 per day | 12 per day |
Total chairs: [latex]20A+30B=620[/latex]
Total tables: [latex]15A+12B=360[/latex]
Of course, using methods from the previous module, Systems of Equations and Inequalities, you can easily solve this system. However every time a new order for chairs and tables comes in, you would have to solve a similar linear system again. It would be better if we had a more general method that could be used to solve the system,[latex]20A+30B=C[/latex] [latex]15A+12B=T[/latex]
in which [latex]C[/latex] and [latex]T[/latex] could be plugged in later. Fortunately, there is a straightforward method for doing exactly that. In this module, you will learn a new method for solving systems, one that uses a table of numbers called a matrix (pl. matrices), very much like the array of numbers from the initial table. The coefficient matrix for this system is:[latex]\begin{bmatrix} {20} & {30} \\[0.3em] {15} & {12} \\[0.3em]
\end{bmatrix}[/latex] Then you can write the system as an equation of matrices, as shown below.[latex]\begin{bmatrix} {20} & {30} \\[0.3em] {15} & {12} \\[0.3em]
\end{bmatrix} \begin{bmatrix} {A} \\[0.3em] {B} \\[0.3em] \end{bmatrix}=\begin{bmatrix} {C} \\[0.3em] {T} \\[0.3em] \end{bmatrix}[/latex] The methods for working with such an equation will be discussed shortly. We’ll revisit our thriving furniture business at the end of the module and see how matrices can be used to solve our table and chair problem once and for all.Learning Objectives
Matrices and Matrix Operations- Find the sum and difference of two matrices
- Find scalar multiples of a matrix
- Find the product of two matrices
Gaussian Elimination
- Write the augmented matrix of a system of equations
- Write the system of equations from an augmented matrix
- Perform row operations on a matrix
- Solve a system of linear equations using matrices
Solve Systems with Inverses
- Find the inverse of a matrix
- Solve a system of linear equations using an inverse matrix
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