Chapter 3 Determinants and Cramer’s Rule
Matrix: an array of numbers
Determinant: a number associated with each matrix.
Consider matrix A:
A= 5
3
 2 1


The determinant of matrix A is notated in the following way:
det a or A
How is this notation different than regular matrix notation?
Determinant of a 2 x 2 matrix:
a
det 
c
b a

d  c
b
=______________________
d
Find the determinant of the below matrices:
1.)
6 2
1 7
2.)
1/ 2 4
2 8
Determinant of a 3 x 3 matrix: ALWAYS MULTIPLY DIAGONALS!!!!!
a
det d
 g
b
e
h
c
f  
i 
Find the determinant of the following matrix:
1.)
Chapter 3 Determinants and Cramer’s Rule
Cramer’s Rule and Systems of Equations
Swiss Mathematician, Gabriel Cramer (1704-1752) developed a way to use matrices to
solve systems of equations. His method has been named “Cramer’s Rule.”
Cramer’s Rule for a 2 x 2 System: (2 equations, 2 variables)
ax  by  c
cd  dy  f
6x  8 y  4
4 x  5 y  4
Let A be the coefficient Matrix of the linear system:
General:

A  

Specific Case:




det A:

A  





det A:
If det A  0 , then the systems has exactly one solution. The solution is:
General:
x
y
Specific Case:
det A
det A
Ordered Pair Solution: _______________________
Chapter 3 Determinants and Cramer’s Rule
Cramer’s Rule for a 3 x 3 System: (3 equations, 3 variables)
General:
ax  by  cz  j
Specific:
dx  ey  fz  k
gx  hy  iz  l
x  2 y  3 z  2
x  y  z  1
3 x  4 y  4 z  4
Coefficient Matrix:
Coefficient Matrix:
Determinant from Calculator: _____
Let A be the coefficient matrix of the linear system. If det A  0 , then the system has
exactly one solution. The solution is:
General:
Specific:
x
y
z
det A
det A
det A
Ordered triple Solution:_____________________
Chapter 3 Determinants and Cramer’s Rule
Use Cramer’s Rule to solve the linear system. Show your setup, but you
may do the work on your calculator.
1.
2x  y  3
5x  6 y  4
x  2y  z  9
2. x  y  z  3
5 x  2 z  1
Word Problem:
Science Connection:
The atomic weights of three compounds are shown. Use a linear system
and Cramer’s rule to find the atomic weights of carbon (C) , and
hydrogen ( H ), and oxygen ( O ).
HW: Determinants of Matrices: pg. 218-219 #12-20 even, 22, 25
Cramer’s Rule: pg. 218-219 #39, 41, 51,52, 60
Chapter 3 Determinants and Cramer’s Rule
To Do on Graphing Calc:
1.) Hit 2nd x 1 (matrix)
Scroll Over to Edit
7.) Hit enter again
2.) Hit Enter and Enter the 2X2
8.) Verify the 2 X2 by hand
3.) Now hit 2nd Quit to get back to the
home screen
4.) Hit 2nd x 1 (matrix)
Scroll Over to math
Highlight 1:det (
5.) Hit enter
6.) To get the matrix into home screen
hit 2nd x 1 (matrix)
highlight 1: [A] and enter
Chapter 3 Determinants and Cramer’s Rule