374 Note for Chapter 4: How to find the determinant of 2 x 2 or 3 x 3 matrices (by hand or with Mathematica)

2 x 2 matrix A =

a
b
c
d

.

A={{a,b},{c,d}}
Out[]=
{{a,b},{c,d}}
Here' s how to make A look more like a matrix we are used to seeing (not required to compute the determinant of matrix A):
MatrixForm[A]
Out[]//MatrixForm=
a
b
c
d
Find the determinant of A. Notation det(A) or |A|.
Det[A]
Out[]=
-bc+ad
Note that
det(A)=|A|=ad-bc
.

3 x 3 matrix B =
a
b
c
d
e
f
g
h
i

B={{a,b,c},{d,e,f},{g,h,i}}
Out[]=
{{a,b,c},{d,e,f},{g,h,i}}
Here's how to make B look more like a matrix we are used to seeing (not required to compute the determinant of matrix B):
MatrixForm[B]
Out[]//MatrixForm=
a
b
c
d
e
f
g
h
i
Det[B]
Out[]=
-ceg+bfg+cdh-afh-bdi+aei
Note that
det(A)=|A|=aei+bfg+cdh-gec-hfa-idb.

Determinants of upper or lower triangular matrices of the form

a
b
0
d

,

a
0
c
d

,
a
b
c
0
e
f
0
0
i
, or
a
0
0
d
e
0
g
h
i
are the product of diagonal elements!

Det[{{a,b},{0,d}}]
Out[]=
ad
Det[{{a,0},{c,d}}]
Out[]=
ad
Det[{{a,b,c},{0,e,f},{0,0,i}}]
Out[]=
aei
Det[{{a,0,0},{d,e,0},{g,h,i}}]
Out[]=
aei