MCV4U Lessons
U8L9: Sec. 8.9.9 Using Matrices to Solve Systems of Equations
We often encounter systems of equations in which the context is not lines and planes. Economists often have to work with systems of dozens of equations involving dozens of unknowns. Relying on a geometric interpretation would not be very useful.
To help organize such vast amounts of data, mathematicians have created a powerful tool called a . A matrix is simply numerical data arranged in a rectangular array. We usually enclose the array in square brackets.
The method of solving a linear system amounts to combining linear combinations of the equations in certain ways. Since several similar steps are involved, this method is ideally suited for technology. Solving a linear system using technology requires a systematic approach because a calculator or computer uses the same method every time.
Elementary Row Operations
A system of linear equations can be represented by a matrix. To obtain an equivalent system, perform any of these operations.
A system of three linear equations in x, y, and z represents three planes in R3. It can be represented by a matrix having the form; , called the where each * represents a real number. When we solve the system using matrices, we attempt to use the to obtain a matrix having the form called the . The method of doing this is called . The matrix that results is the reduced matrix and is in
Example # 1: (Solving a Matrix in R2)
Solve the linear system using matrices.
Solution:
Example # 2: (Solving a Matrix in R3)
Solve the linear system using matrices
Solution:
Homework: Handout 8.9.9
U8L10: Sec. 8.9.10 Matrices Continued……
It is not always possible to reduce a matrix of the form to one of the form using row reduction. If one of the equations is a linear combination of the other two, a will occur at some point. It may be possible to reduce the matrix to a form such as . Then the system is , and the corresponding planes intersect in a line. Parametric equations of this line constitute the solution of the system.
Example # 1:
Solve the system using matrices and interpret the solution geometrically.
Solution:
Example # 2:
Solve the following system of linear equations by reducing the corresponding augmented matrix to row-reduced echelon form.
Solution:
Handout (Representations of Solutions to Systems of Equations in R3)
Homework: Handout 8.9.10
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