The Determinant of Square Matrix

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  • Created by Deaw JAIBUN, Mahidol Wittayanusorn School

    fWhat you should learn -

    2x2 -

    -

    fWhy you should learn it

    2. The Determinant of Square Matrix

    (determinant) n nn A (scale factor) A (multilinear algebra) (substitution rule) n nn (commutative ring)

    1. 2. A A det(A) |A|

    2 X 2 2 x 2

    d

    a bA

    c

    =

    A

    det( ) |A| a b

    A ad bcc d

    = = =

    3 X 3 1 2 2

    A 3 x 3 A = 1 1 1

    2 2 3

    3 3 3

    a b c

    a b c

    a b c

    A det (A) 1 1 1 1 12 2 3 2 2

    3 3 3 3 3

    a b c a b

    a b c a b

    a b c a b

    1 2 3 1 2 3 1 2 3 3 2 1 3 2 1 3 2 1

    det( ) ( ) ( )A a b c b c a c a b a b c b c a c a b= + + + +

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    1 A = [ 5 ] B = [ -8 ] det( )A det( )B

    2 4 3 4 2 1 5, ,2 1 2 1 2 1

    A B C = = =

    det( )det( )det( )A B C

    3 1 2 3

    0 4 1

    1 2 0

    A

    = det( )A

    4 2 1 0

    3 4 5

    9 8 7

    A

    = det( )A

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    2.1 Minor Cofactor 2.1.1 Minor A n x n n > 2 1.

    ija

    ijM

    2. ija i j

    A

    5 2 1 0

    3 4 5

    9 8 7

    A =

    A

    2.1.2 Cofactor A n x n n > 2 1.

    ija

    ijC

    2. ija ( 1)i j

    ijM+

    6 2 0 1

    3 1 2

    4 5 6

    A

    =

    A

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    Minor Cofactor 2.1.1 [ ]

    ij n nA a = ija n > 2

    1. 1 1 2 2

    det( ) ... ,i i i i in in

    A a C a C a C i= + + + i 2.

    1 1 2 2det( ) ... ,

    j j j j nj njA a C a C a C j= + + + j

    7 A = 4 1 12 1 2

    3 5 2

    det (A)

    8 A = 1 1 1 2

    1 2 2 1

    4 3 0 1

    3 0 2 1

    det (A)

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    2.1.3 [ ]

    ij n nA a = ijC ija

    A (adjoint of A) ( )adj A

    11 12 1 11 21 1

    21 22 2 12 22 2

    1 2 1 2

    ( )

    T

    n n

    n n

    n n nn n n nn

    C C C C C C

    C C C C C Cadj A

    C C C C C C

    = =

    " "" "

    # # # # # #" "

    2.1.2 [ ]

    ij n nA a = ija , 2n n

    1. ( ) ( ) det( ) Aadj A adj A A A I= = 2. det( ) 0A 1 1 ( )

    det( )A adj A

    A =

    Proof

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    9 A A = 1 0 12 1 0

    1 1 1

    10 3 2 1

    5 6 2

    1 0 3

    A

    =

    11 12 131

    21 22 23

    31 32 33

    b b b

    A b b b

    b b b

    =

    31 23b b+

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    fWhat you should learn -

    -

    - ERO

    (Elementary Row Operation)

    2.2.2 A n x n det (A) 0 AX = B x1 , x2 , , xn b1 , b2 , ,bn

    X = 1

    2

    n

    x

    x

    x

    # , B =

    1

    2

    n

    b

    b

    b

    #

    x1 = 1det( )

    det( )

    A

    A , x2 = 2

    det( )

    det( )

    A

    A, , xn = det( )

    det( )nA

    A

    Ai i A B

    11 2 3 9

    2 3 3

    x y

    x y

    + = =

    12

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    3 3

    2 3 20

    7 23

    x y z

    x y z

    x y z

    + = + = + + =

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    2.1

    1. 1 2 3

    6 7 1

    3 1 4

    A

    =

    a. A b. A c. det(A) d. A-1

    2.

    4 0 0 1 0

    3 3 3 1 0

    1 2 4 2 3

    9 4 6 2 3

    2 2 4 2 3

    A

    =

    1A

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    3-5 3. 3 2 13

    3 2 5

    x y

    y

    + = =

    4. 2 4 5

    2 2 19

    x y

    x y

    + = + =

    5. 2 3 8

    4 2 4

    3 2 9

    x y z

    x y z

    x y z

    + = + + = + =

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    2.2 2.2.1 A A det( ) 0A = Proof 2.2.2 A det( ) det( )TA A= Proof 2.2.3 A 1. () A c

    c det (A) 2. ( ) A

    - det (A) 3. () A

    () A () det(A)

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    2.2.4 A (Elementary Matrix) 1. E

    nI k det( )E k=

    2. E nI det( ) 1E =

    3. E nI

    det( ) 1E =

    13 0 1 5

    3 6 9

    2 6 1

    A

    =

    14 1 0 0 3

    2 7 0 6

    0 6 3 0

    7 3 1 5

    A

    =

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    2.2 1. det( ) det( )TA A=

    a. 2 31 4

    A =

    b. 1 2 3

    6 7 1

    3 1 4

    A

    =

    2.

    a. 3 17 4

    0 5 1

    0 0 2

    b. 2 0 0 0

    8 2 0 0

    7 0 1 0

    9 5 6 1

    c. 1 0 0 0

    0 1 0 0

    0 0 5 0

    0 0 0 1

    d. 1 0 0 0

    0 0 1 0

    0 1 0 0

    0 0 0 1

    e. 1 0 0 0

    0 1 0 9

    0 0 1 0

    0 0 0 1

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    3. a. 1 2 3

    6 7 1

    3 1 4

    A

    =

    b. 2 1 3 11 0 1 10 2 1 0

    0 1 2 3

    A

    =

    4. 6a b c

    d e f

    g h i

    =

    a. d e f

    g h i

    a b c

    b. 3 3 3

    4 4 4

    a b c

    d e f

    g h i

    c. 3 3 3

    4 4 4

    a b c

    d e f

    g d h e i f

    5. 2.2.4

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    2.3 2.3.1 A, B n x n k 1. det(kA) = kndet(A) 2. det(Ak) = [det(A)]k k

    2.3.2 A, B C n x n r r C r A B det(C) = det(A) + det(B)

  • | Page 16

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    2.3.3 A n x n E n x n det(EA) = det(E) det(A)

    2.3.4 A det(A) 0

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    2.3.5 A,B n x n det(AB) = det(A) det(B)

    2.3.6 A 1 1det( )

    det( )A

    A =

    2.3.7 A n x n

    1. A 2. 3.

    RR nA I=

    4. A 5. AX B=

    B 1n 6. AX B=

    B 1n 7. det(A) 0

    Note: 1. A = B det (A) = det (B) 2. det (A) = det (B) A = B 3. det (A+B) det (A) + det (B)

  • | Page 18

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    15 det(A) A = 1 1 1 2

    1 2 2 1

    4 3 0 1

    3 0 2 1

    16 det(A) A = 1 1 3 1

    1 1 1 2

    3 2 2 1

    2 3 2 5

    17 A = 3 2

    2 1 2

    0 3 2

    x

    x

    B = 2

    3 1

    x x

    x det(A) = 4 det(B)

  • | Page 19

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    18 A = 1 2 2 1 2 1 2 1, , ,3 4 3 0 4 3 4 3

    A B C D = = = =

    det(ABCD)

    19 2 1 5 3 3 4 3 41 1 1 1 2 1 2 1

    a b

    c d

    =

    a bc d

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    A =x x

    A =x x

    ( )n

    n

    A

    A I

    A I

    = = =

    x x 0x x 0

    x 0

    ( )n

    A I =x 0 () ( )

    nA I =x 0

    ( ) (nontrivial solutions) (eigenvalue) A (eigenvector) A ( )

    nA I =x 0

    nA I

    nA I ( )

    nA I =x 0

    det( ) 0n

    A I =

    20 1 2 1

    1 2 2

    3

    4 2

    x x x

    x x x

    + = + =

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    2.3 1. det(kA)

    a. 1 2 , 23 4

    A k = =

    b. 2 1 3

    3 2 1 , 2

    1 4 5

    A k

    = =

    2. det(AB) 2 1 0 1 1 3

    3 4 0 , 7 1 2

    0 0 2 5 0 1

    A B

    = =

    3. det(A)=0 2 8 1 4

    3 2 5 1

    1 10 6 5

    4 6 4 3

    A

    =

    4. k

    a. 3 22 2

    kA

    k

    =

    b. 1 2 4

    3 1 6

    3 2

    A

    k

    =

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    5. i. ii. iii.

    a. 1 2 1

    1 2 2

    2

    2

    x x x

    x x x

    + = + =

    b. 1 2 11 2 2

    2 3

    4 3

    x x x

    x x x

    + = + =