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Solving Systems of Linear Equations Using Inverse Matrices

For questions 333 and 334, solve using inverse matrices. Refer to the following

discussion, as needed.

A solution to the linear system is an ordered pair of numbers (s1

,

s2) such that are both true. When the system has a solution, it is

consistent; otherwise, the system is inconsistent. By the definition of matrix

multiplication, the linear system is equivalent to the matrix

equation . In matrix notation this equation has the form AX = C,

where A is the 2 × 2 coefficient matrix , X is the 2 × 1 column vector of

variables , and C is the 2 × 1 column vector of constants . If A–1 exists,

then the system has the unique solution X = A–1 C. Caution: If A does not have an

inverse, then X = A–1C does not apply.

Note: This method of solving systems of linear equations works for any n × n

system of linear equations, provided the coefficient matrix has an inverse. Still,

for systems larger than 2 × 2, the computations can be long and tedious, so using

technological aids to perform the calculations is your best strategy.

333.

334.

Solving Systems of Linear Equations Using Cramer’s Rule

Overview of Cramer’s rule:

The 2 × 2 system of linear equations , where is the

coefficient matrix, is the column vector of variables, and is the

column vector of constants, has the unique solution: ,

provided that ≠ 0. Let Xi be the determinant of the matrix obtained by

replacing the ith column of A with the column vector K and let D be , the

determinant of the coefficient matrix, then the solution can be written as

.

Similarly, Cramer’s rule applied to a 3 × 3 system of linear equations

yields the unique solution:

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