Determinants
• All square matrices have a determinant. The only ones we will
deal with are 2x2 and 3x3 matrices.
• To find the determinant of a 2x2 matrix, you use the rule for
second order determinants.
•
ad-bc
• There are two ways to find the determinant of a 3x3 matrix.
• First is expansion by minors. To get your minors, pick any
row of the matrix look at the first digit, it will become a multiple
for a 2x2 matrix. Ignore the other numbers in the row and cover
up the other numbers in the column. You will only have 4
numbers left in the matrix and take that as a 2x2 matrix. Use
the digit you started with as a multiple of the matrix. Repeat
those steps for the remaining two digits in the row. Look at the
next page for the formula.
Determinants
• This is assuming you pick the first row for your multiples but
remember that you can pick any row.
= a*
-b*
+c*
• Observe how when a is the multiple, the corresponding matrix is
only the elements left over when the row and column of a are
covered.
• The other way to find the determinant of a 3x3 matrix is by using
diagonals. Look at the next page to see how it works.
Determinants
• First take the first two columns and add them on to the end.
• Then draw diagonals from the first entry of each row down and to
the right. You obtain aef, bfg, and cdh. Then start at the bottom
and draw diagonals up and to the right. You get gec, hfa, and idb.
• The determinant will equal aef+bfg+cdh-gec-hfa-idb.
Problems
Solve the following:
1)
2)
3)
Solve by expansion by minors:
4)
5)
6)
8)
9)
Solve by diagonals:
7)
• Answers:
• 1) 14 2)-22
3)-22
4)-32
5)95 6) 369
7)-323
8)-30
9) 90