, scalar d is called A matrix ) if there is a nonzero vector (eigenvector ) Ax A X
'
a an eigenvalue of (n x n x such that =
the collection of all
eigenvectors corresponding to d
together with the zero vector, is called the
eigen space of d Ea
'
:
,
'
determinant of a 2×2 matrix A =
(da! %!) ,
det A =/ Al =
a. zazz a.zaz ,
-
[aag aa÷ aaa};] aa: I:} % I:3 % aa!!
A '
ll
'
determinant of a 3×3 matrix A = aeta -
- lat = a
-
a ta
.. , , ,
.. . ,
, ,
" t 't
( i, j) cofactor of A
Cig =L 1) A. ij
'
-
.
-
det
÷÷ '÷÷ '÷ I
-
check it cofactor is + or -
. .
.
-
the Laplace Expansion Theorem the determinant of an matrix A [a ;j] 22 det A C taizcizt tain Cin
'
: n x n =
,
where n = a . . .
;, ;,
n
E ( along i'throw
aijcij
= )
j= I
det A = a . C . t a C t ta C
zj 2J
. . .
n'j y nj nj
E (along j
'
aijcij
= th
i = I
column )
the determinant of a
triangular matrix det A
'
=
a
Azz ann
. . -
,,
-
A =
[a ;j ] is a square matrix a . if A has a zero row ( column ) det A = o
b . if B is obtained by interchanging two rows ( columns ) of A det B = -
det A
C .
if A has two identical rows ( columns ) det A = o
d . if B is obtained by multiplying a row ( column ) of A by K det B = K det (A )
e . if A B , C are identical except that the ith
,
row (column ) of C is the sum of the ith rows
(columns ) of A and B det C = det At det B
t . if B is obtained by adding a multiple of one row ( column ) of A to another row ( column )
det B = det A
'
E is an nxn
elementary matrix a . if E results from interchanging two rows of In det E = -
I
b . if E results from multiplying one row of In by K det E = K
C . if E results from adding a multiple of one row of In to another row det E =L
B is matrix and E is an matrix ( EB ) ( det E ) ( det
'
an n x n n x n
elementary det = B)
a square matrix is invertible iff det A t o
'
-
if A is an nxn matrix det ( KA ) = kndet A
if A and B are n x n matrices det ( AB ) =
( det A) (det B)
-
I
'
-
if A is invertible det ( A- ) = det A
for matrix det A aet A
any square
-
=
'
a an eigenvalue of (n x n x such that =
the collection of all
eigenvectors corresponding to d
together with the zero vector, is called the
eigen space of d Ea
'
:
,
'
determinant of a 2×2 matrix A =
(da! %!) ,
det A =/ Al =
a. zazz a.zaz ,
-
[aag aa÷ aaa};] aa: I:} % I:3 % aa!!
A '
ll
'
determinant of a 3×3 matrix A = aeta -
- lat = a
-
a ta
.. , , ,
.. . ,
, ,
" t 't
( i, j) cofactor of A
Cig =L 1) A. ij
'
-
.
-
det
÷÷ '÷÷ '÷ I
-
check it cofactor is + or -
. .
.
-
the Laplace Expansion Theorem the determinant of an matrix A [a ;j] 22 det A C taizcizt tain Cin
'
: n x n =
,
where n = a . . .
;, ;,
n
E ( along i'throw
aijcij
= )
j= I
det A = a . C . t a C t ta C
zj 2J
. . .
n'j y nj nj
E (along j
'
aijcij
= th
i = I
column )
the determinant of a
triangular matrix det A
'
=
a
Azz ann
. . -
,,
-
A =
[a ;j ] is a square matrix a . if A has a zero row ( column ) det A = o
b . if B is obtained by interchanging two rows ( columns ) of A det B = -
det A
C .
if A has two identical rows ( columns ) det A = o
d . if B is obtained by multiplying a row ( column ) of A by K det B = K det (A )
e . if A B , C are identical except that the ith
,
row (column ) of C is the sum of the ith rows
(columns ) of A and B det C = det At det B
t . if B is obtained by adding a multiple of one row ( column ) of A to another row ( column )
det B = det A
'
E is an nxn
elementary matrix a . if E results from interchanging two rows of In det E = -
I
b . if E results from multiplying one row of In by K det E = K
C . if E results from adding a multiple of one row of In to another row det E =L
B is matrix and E is an matrix ( EB ) ( det E ) ( det
'
an n x n n x n
elementary det = B)
a square matrix is invertible iff det A t o
'
-
if A is an nxn matrix det ( KA ) = kndet A
if A and B are n x n matrices det ( AB ) =
( det A) (det B)
-
I
'
-
if A is invertible det ( A- ) = det A
for matrix det A aet A
any square
-
=
