Systems of Linear Equations and Matrices

Disclaimer: The concepts to know on the sheet is not necessarily all inclusive, but it should be a guide of
main concepts to know and kinds of problems to be able to solve as you study for the test.

Section 2.1: Systems of Linear Equations
• Be able to solve systems of linear equations (2 equations and 2 variables)
algebraically using Method of Elimination and Substitution Method .
• Be able to solve system of linear equations graphically (remember you need to
write down the equations you plugged into your calculator, sketch the graph and
label the solution)
• Understand the different types of solutions for a system of 2 equations and 2
variables that can be obtained algebraically (one solution, no solution, infinitely
many solutions) and how it correlates to the graphically solution (lines intercept at
one point, lines are parallel, or same line twice)
• Know what it means when a system of linear equations is said to be consistent,
inconsistent, independent or dependent.

Section 2.2: Using Matrices to Solve systems of linear equations.
• Understand what is a matrix and be able to determine the dimension (i.e. size) of a
matrix
• Be able to write a system of linear equations as an augmented matrix
• Understand what it means for a matrix to be in RREF ( Reduced Row Echelon
Form) and be able to determine if a matrix is in RREF or not.
• Be able to solve a system of linear equations using Gaussian Elimination . You
need to be able to solve a system algebraically showing all Row Operations and
get the matrix in RREF to obtain the solution.
• Understand how to interpret a matrix in RREF to determine the solution of the
system (remember a system can have one solution, no solutions, or infinitely
many solutions). If you have an infinite number of solutions for the system, be
able to give the solutions in terms of one of the variables and list some solutions
to the system.

Section 2.3: Applications involving Systems of Linear Equations
• Use calculator to get an augmented matrix in RREF.
• Given an application problem, be able to set up a system of linear equations and
solve the system to get the solution. Remember to identify your variables
completely (i.e. let x =number of tickets in Upper Level region etc) and use units
for the answers.

Section 3.1: Matrix Addition and Scalar Multiplication
• Be able to add and subtract matrices (and know when matrix addition or
subtraction is undefined)
• Be able to perform scalar multiplication on a matrix

Section 3.2: Matrix multiplication and Inverses
• Be able to perform matrix multiplication (and know when matrix multiplication is
undefined).
• Understand what is an Identity Matrix.
• Understand what an inverse matrix is and given 2 matrices, be able to show one
matrix is the inverse matrix of the other.
• For a 2x2 matrix, be able to find the inverse matrix using a formula. Note: On a
quiz or exam you will not be asked to find the inverse matrix of a 2x2 matrix
using the formula from this section. However you may want to know the formula
as another method for finding the inverse matrix of a 2x2 matrix if you choose so.
• Understand what kinds of matrices have an inverse matrix.
• Understand what it means for a matrix to be Singular and Invertible.

Section 3.3: Solving Matrix Equations (Using the Inverse Matrix to Solve Systems of
Linear Equations)
• Be able to find the inverse matrix algebraically (or determine algebraically if an
inverse matrix does not exist) of any square size matrix using the row operations
(i.e. set up augmented matrix and use row operations to get matrix in RREF)
• Be able to use the graphing calculator to find an inverse matrix (or determine if an
inverse matrix does not exist). Give inverse matrix using exact values and not
approximations (convert repeating decimals to fraction form )
• Be able to write a system of linear equations (those with the same number of
equations as variables) as a product of matrices and write as a matrix equation
AX=B. Using matrix “A”, be able to solve a system of equations using the inverse
matrix of A, (i.e. A⁻¹ ) and solve the system of equations using the inverse matrix
and matrix multiplication.
• Understand the restrictions of using the inverse matrix to solve systems of linear
equations (only works on systems that are independent (i.e. have a unique
solution). You will need to use other algebraic methods to determine if the
system is dependent (infinitely many solutions) or inconsistent (no solutions)

On-Line Section: Cramer’s Rule:
• Be able to calculate the determinant of a 2x2 matrix
• Be able to use Cramer’s rule to solve a system of 2 equations and 2 variables.
• Understand the restrictions of using Cramer’s rule (get determinant of zero in the
denominator ) to solve a system of linear equations (only can solve systems that
are independent - i.e. have a unique solution. Need to use other algebraic methods
to determine if the system is dependent (infinitely many solutions) or inconsistent
(no solutions).

Section 4.1: Graphing Linear Inequalities
• Given a linear inequality or a system of linear inequalities, be able to graph the
solution region (make sure you can find the x and y intercepts (i.e. horizontal and
vertical intercepts algebraically)
• Be able to determine all corner points of the solution region (be able to
algebraically find the corner points that are not intercepts)

Section 4.2: Solving Linear Programming Problems Graphically.
• Given a linear programming problem, be able to find the optimal solution and
optimal value. You will need to be able to graph the constraints and determine
the feasible region, find all corner point to the feasible region and use the
objective function to find the optimal solution (i.e. the maximum or minimum
value). Understand cases where an optimal solution may not exist (If the feasible
region is unbounded). You will need to be able to find the x and y intercepts
(horizontal and vertical intercepts) algebraically and find corner points that are
not intercepts algebraically.
• Given an application problem be able to identify the variables, and come up with
the objective function and all the constraints, and solve the linear programming
problem.

Summary of methods for solving system of linear equations:
Given any system of 2 equations and 2 variables you should be able to solve using the
following methods:
• Graphically
• Method of Elimination
• Substitution Method
• Gaussian Elimination (algebraically and using calculator – rref feature)
• Inverse Matrix method (be able to find inverse matrix algebraically and by
calculator)
• Cramer’s Rule

Given any system of 3 equations and 3 variables you should be able to solve using
the following methods:
• Gaussian Elimination (algebraically and using calculator – rref feature)
• Inverse Matrix method (be able to find inverse matrix algebraically and by
calculator