8.7. Determinant of Matrices http://www.ck12.org
The smaller 2×2 matrices are the entries that remain when the row and column of the coefficient you are working
with are ignored.
detB=|B|= +a·
∣∣
∣∣he fi∣∣
∣∣−b·∣∣
∣∣dg fi∣∣
∣∣+c·∣∣
∣∣d eg h∣∣
∣∣
Next take the determinant of the smaller 2×2 matrices and you get a long string of computations.
= +a(ei−f h)−b(di−f g)+c(dh−eg)
=aei−a f h−bdi+b f g+cdh−ceg
=aei+b f g+cdh−ceg−a f h−bdiMost people do not remember this sequence. A French mathematician named Sarrus demonstrated a great device
to memorize the computation of the determinant for 3×3 matrices. The first step is simply to copy the first two
columns to the right of the matrix. Then draw three diagonal lines going down and to the right.
B=
a b c
d e f
g h i
Notice that they correspond exactly to the three positive terms of the determinant demonstrated above. Next draw
three diagonals going up and to the right.These diagonals correspond exactly to the three negative terms.
detB=aei+b f g+cdh−ceg−a f h−bdi
Sarrus’s rule does not work for the determinants of matrices that are not of order 3×3.
Example A
Find detAforA=
[3 2
1 5
]
Solution:
∣∣
∣∣3 21 5
∣∣
∣∣= 3 · 5 − 2 · 1 = 15 − 2 = 13
Example B
Find detBforB=
3 2 1
5 0 2
2 1 5
Solution:
∣∣
∣∣
∣∣