Download Lesson 1.7: Solving Systems with Matrix Equations

Lesson 1.7: Solving Systems with Matrix Equations
A system of linear equations can be written as a
of the form
.
Once the equations have been expressed in standard form, matrix A represents the coefficients of the
variables, matrix X represents the variables themselves, and matrix B represents the constants on the
right-hand sides of the equations.
2 y  z  7  5 x

e.g. a) Write the system  x  2 y  2 z  0 as a matrix equation.
3 y  17  z

Solving a matrix equation of the form AX  B is similar to solving a linear equation of the form ax  b .
Just as
Real Numbers
Matrices
ax  b
1
1
 ax   b
a
a
1
1 x   b
a
1
x  b
a
AX  B
A1 AX  A1B
IX  A1B
X  A1B
1
must exist in order to solve ax  b (meaning that a  0 ), A1 must exist to solve AX  B
a
(meaning that
).
When solving the matrix equation AX  B , it is absolutely critical that we PRE-multiply both sides of
the equation by A1 (meaning that we should calculate A1 B and NOT BA1 ), because
.
e.g. b) Solve the matrix equation from e.g. a)
As you already know, not all systems of linear equations have solutions. In a matrix equation of the
form AX  B , if the coefficient matrix A is singular (i.e. has no inverse), then the system represented
by the matrix equation does not have a unique solution, meaning that it will either be an inconsistent
system with no solutions, or a consistent dependent solution with infinitely many solutions.
3x  4 y  3
, if possible, by using a matrix equation. If it is not possible,
6 x  8 y  18
e.g. c) Solve 
classify the system.
Matrix equations are a much more efficient method than elimination or substitution when solving large
systems of equations. For that matter, it is probably best to use matrix equations anytime the system is
3  3 or larger.