DETERMINANT
definition
and
origin
Associated with every square matrix A there is
a number called the determinant of A.
The definition of the determinant of a 22 matrix follows:
det
(
a c
b d
)
= ad - bc
Associated with every square matrix A there is
a number called the determinant of A.
The definition of the determinant of a 22 matrix follows:
det
(
a c
b d
)
= ad - bc
Associated with every square matrix A there is
a number called the determinant of A.
The definition of the determinant of a 22 matrix follows:
det
(
a c
b d
)
= ad - bc
The concept “determinant” arose from
early attempts to generalize the process
of solving systems of linear equations.
Consider the following:
Solve the system :
ax
 cy
 j
bx  dy  k
Solve the system :
a c 
  ad  bc
det 
b d 
ax
 cy
 j
bx  dy  k
a j
  ak  bj
det
b k 
 j c
  dj  ck
det
k d 
Solve the system :
a c 
  ad  bc
det 
b d 
a c 


b d 
ax
replace y
coordinates
with j
k
 j
bx  dy  k
a j
  ak  bj
det
b k 
a j


b k 
 cy
 j c
  dj  ck
det
k d 
 j c


k d 
replace x
coordinates
with j
k
Solve the system :
a c 
  ad  bc
det 
b d 
ax
 cy
 j
bx  dy  k
a j
  ak  bj
det
b k 
 j c
  dj  ck
det
k d 
Solve the system :
a c 
  ad  bc
det 
b d 
ax
 cy
 j
bx  dy  k
ax
 j
bx  dy  k
a j
  ak  bj
det
b k 
to solve for
 cy
y:
 j c
  dj  ck
det
k d 
 b(ax
a(bx
 cy
 j)
 dy  k )
ax
Solve the system :
a c 
  ad  bc
det 
b d 
ax
 cy
 j
bx  dy  k
 cy
bx  dy  k
a j
  ak  bj
det
b k 
to solve for
y:
 j c
  dj  ck
det
k d 
 b(ax
a(bx
 abx  bcy  bj
abx
 ady
 j
 ak
(ad  bc) y  ak  bj
ak  bj
y
ad  bc
 cy
 j)
 dy  k )
ax
Solve the system :
a c 
  ad  bc
det 
b d 
ax
 cy
 j
bx  dy  k
 cy
bx  dy  k
a j
  ak  bj
det
b k 
to solve for
y:
 j c
  dj  ck
det
k d 
 b(ax
a(bx
 abx  bcy  bj
abx
 ady
 j
 ak
(ad  bc) y  ak  bj
ak  bj
y
ad  bc
 cy
 j)
 dy  k )
Solve the system :
a c 
  ad  bc
det 
b d 
ax
 cy
 j
bx  dy  k
ax
 j
bx  dy  k
a j
  ak  bj
det
b k 
to solve for
 cy
x:
 j c
  dj  ck
det
k d 
d (ax
 cy
 j)
 c(bx  dy  k )
ax
Solve the system :
a c 
  ad  bc
det 
b d 
ax
 cy
 j
bx  dy  k
a j
  ak  bj
det
b k 
 cdy
 j
bx  dy  k
to solve for
adx
 cy
x:
 j c
  dj  ck
det
k d 
d (ax
 cy
 j)
 c(bx  dy  k )
 dj
 bcx  cdy  ck
(ad  bc) x  dj  ck
dj  ck
y
ad  bc
ax
Solve the system :
a c 
  ad  bc
det 
b d 
ax
 cy
 j
bx  dy  k
a j
  ak  bj
det
b k 
 cdy
 j
bx  dy  k
to solve for
adx
 cy
x:
 j c
  dj  ck
det
k d 
d (ax
 cy
 j)
 c(bx  dy  k )
 dj
 bcx  cdy  ck
(ad  bc) x  dj  ck
dj  ck
y
ad  bc
The following presentation entitled
will outline the algorithm for evaluating
the determinant of a 33 matrix