Download Notes: Matrix Equations

Matrix Equations
● Step 1: Write the system as a matrix
equation. A three-equation system is shown
below.
a1 x  b1 y  c1z  C1
a2 x  b2 y  c2 z  C2
a3 x  b3 y  c3 z  C 3
 a1
a
 2
 a3
b1
b2
b3
c1   x   C1 
c2   y   C2 
c3   z   C3 
Matrix Equations
● Step 2: Find the inverse of the coefficient
matrix.
Note: This can be done easily for a 2 x 2
matrix. For larger matrices, use a calculator
to find the inverse.
Matrix Equations
● Step 3: Multiply both sides of the matrix
equation by the inverse.
The inverse of the coefficient matrix times the
coefficient matrix equals the identity matrix.
1
 x   a1 b1 c1   C1 
 y    a b c   C 
   2 2 2  2
 z   a3 b3 c3   C3 
Note: The multiplication order on the right side is
very important. We cannot multiply a 3 x 1 times a
3 x 3 matrix!
Matrix Equations
● Example: Solve the system
 3 2   x   9 
3x - 2y = 9
1 2   y    5 

   
x + 2y = -5
1
 3 2 
1  2 2
1 2   8  1 3




x 1  2 2  9 
 y   8  1 3   5 
 

 
Matrix Equations
● Example, continued:
x 1  2 2  9 
 y   8  1 3   5 
 

 
Multiply the matrices (a ‘2 x 2’ times a
‘2 x 1’) first, then distribute the scalar.
x 1  8 
 y   8  24 
 


x  1 
 y    3
   
Matrix Equations
• Example #2: Solve the 3 x 3 system
 3 2 1   x   9 
3x - 2y + z = 9
1 2 2   y    5 
x + 2y - 2z = -5

   
x + y - 4z = -2
1 1 4   z   2 
Using a graphing calculator,
1
 236
 3 2 1 
1 2 2     2
 23


 231
1 1 4 
7
23
13
23
5
23
 232 
7 
 23 
 238 
Matrix Equations
● Example #2, continued
 x   236
 y    2
   23
 z   231
 x  1 
 y    3
   
 z   0 
7
23
13
23
5
23
 232 
7 
 23 
 238 
9
 5
 
 2