3x3 matrices
IB SL/HL maths
www.ibmaths.com
3x3 Matrices
By the end of this lesson you will be able to:
•
•
•
find the determinant of a 3x3 matrix
without a GDC.
find inverses and determinant of 3x3
matrices using a GDC.
solve simultaneous equations using in
3 or more unknowns using a GDC.
Determinant of a 3x3 matrix without a GDC
a b c


de td e f  a(ei  fh)  b(di  fg)  c(dh  eg)
g h i 


Sometimes this is better seen and remembered by using a
diagram:

-
+
Highlight a, then cancel the numbers in a’s column and row.
Find the determinant of the 2x2 matrix remaining.
Continue for b and c. Remember to subtract b.
Now some practise…
Find the determinants of each matrix.
1 2 3 


A  2 0 2
7 3 5


8 1 9 


B  3 0 1
2 3 1 


A  60
B  104

1 3 4 


C  2 2 1
1 2 3 


C  41
2 1 1


A  2 0 1 
1 3 2


Using your GDC to get the
determinant.
The TI
1.Enter a 3x3 matrix.
Go to the Matrix function by
2nd x-1.
Tab across to Edit, and ENTER.
Choose the dimensions: 3x3, and
enter in your matrix.
2. Quit when you have entered
your matrix. Go back into the
Matrix function, and tab across
to MATH, choose option 1; now
go back to Matrix and choose A.
The Casio
1. From the main menu choose
Matrix option, choose a
 matrix and set it’s
dimensions.
Enter the matrix.
2. Go back to the main menu
and go into Run.
3. Choose OPTN, F2, F3
(Det), then F1 (Mat),
ALPHA, and choose the
matrix you entered.
Using your GDC to get the inverse
of a 3x3 matrix.
2 1 1


A  2 0 1 
1 3 2


The TI
1. Enter the matrix as before.
2. Go into the Matrix menu and
select matrix from the first
menu.
3. Now select the x-1 and ENTER.
4. To get the numbers as
fractions you must now enter
ANS FRAC (from the MATH
menu).
The Casio
1. Enter the matrix as before.

2. Go into the Run menu, and
choose OPTN, followed by F2
and F1, ALPHA and choose
the matrix.
3. When the matrix is on the
screen put it to the power of
negative 1.
Ensure that you use SHIFT ) and
not ^ button.
2 1 1


A  2 0 1 
1 3 2


Solving a simultaneous equation with
3 unknowns.
Solve these simultaneous equations:
2x  y  z  5

2x  z  6
x  3y  2z  3
Look at the simultaneous equations
and it could be written as:

2 1 1x 5 

   
2 0 1 y  6
1 3 2z  3

   
This can be written as:

x 5 
   
Ay  6
z  3
   
Remove the A by multiplying
both sides by A-1.

x  1
  
y  1
z  
  2

1
1 
  
3
3 5 
1 0 6
5
2  
 3
3
3 
Now on your GDC multiply
the two matrices to give the
values: x=4, y=-1, z=2
Now solve these equations:
1.
x  2y  3z  11
4x  3y  2z  1
2.


xy z  6
2x  y  z  4
x  2y  z  4
3x  y  2z  6
3.

x  2,y  3, z  1
1
1
5
x  ,y  , z 
2
2
2
x  y  z  2
3x  4y  z  1

2x  5y  2z  13
x  3,y  1, z  6