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Mathematics Dictionary
Dr. K. G. Shih
Matrix and Determinant
Subjects


  • AL 13 00 | - Outlines
  • AL 13 01 | - Menu of Computation Tools
  • AL 13 02 | - Definitions
  • AL 13 03 | - Properties of Matrix
  • AL 13 04 | - Properties of determinant
  • AL 13 05 | - Value of 4th order determinant
  • AL 13 06 | - Value of 3rd order determinant
  • AL 13 07 | -
  • AL 13 08 | -
  • AL 13 09 | -
  • AL 13 10 | - Examples
  • AL 13 11 | - Examples
    • Answers


    AL 13 01. Menu of computation tools

    PC computer Program in MD2002 ZM35
    • Menus
      • 1. Production of matrix
      • 2.
      • 3.
      • 4. Adjoint matrix
      • 5. Inverse matrix by adjoint matrix
      • 6. Inverse matrix by pivot
      • 7. Determinant evaluation
      • 8. Determinant evaluation

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    AL 13 02. Defintions

    Terms of matrix
    • 1. Rectangular Matrix : A rectangular arrys of numbers enclosed by parenthesis
    • 2. Square Matrix : A square arrys of numbers enclosed by parenthesis
    • 3. Elements
      • The individual entries in the matrix.
      • Symbol in computer language : A(i,j) is element at ith column and jth row
    • 4. Column vector : An matrix has m rows and one column
    • 5. Diagonal matrix
      • The elements are sero if the row number is not equal column number
      • That is A(i,j) = 0 if i is not equal to j
    • 6. Unit matrix
      • The elements are sero if the row number is not equal column number
      • The elements are one if row number equals column number
      • Example : a 3rd order unit matrix
        • | 1 0 0 |
        • | 0 1 0 | = I
        • | 0 0 1 |
    • 7. Inverse matrix
      • Two square matrix : Mat[A] and Mat[B]
      • If Mat[A]*Mat[B] = I, the Mat[B] is inverse matrix of Mat[A]
    • 8. Sub matrix
      • It is a new matrix by deleting ith row and jth column of an matrix
      • Example : Sub matrix of a 3rd order matrix by deleting 1s row and 1st column
        • | a22 a32 |
        • | a32 a33 |
    Definition of determinant
    • 1. Square matrix of order n can associate with number which is called determinant
    • 2. Co-factor
      • Co-factor of Det[A] is A(i,j) = {(-1)^(i+j))*Det[B]
      • Det[B] is derterminant of Det[A] without ith row and jth column
    Example : express 3rd order determinat in co-factor form
    • co-factor of 1st row and 1st column
      • | a22 a32 |
      • | a23 a33 | * (-1)^(1+1)
    • co-factor of 1st row and 2nd column
      • | a21 a23 |
      • | a31 a33 | * (-1)^(2+1)
    • co-factor of 1st row and 3rd column
      • | a21 a22 |
      • | a31 a32 | * (-1)^(3+1)

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    Q03. Properties of marix


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    AL 13 04.Properties of determinant

    Value of determinant
    • 1. If two rows are identical, the value of the determinat is zero
    • 2. If two columns are identical, the value of the determinat is zero

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    AL 13 05. Evaluate 4th order determinant

    Co-fector of 1st column and 1st row
    • | a22 a32 a42 |
    • | a23 a33 a43 |*((-1)^(1+1)) = V1
    • | a24 a34 a44 |
    Co-fector of 2nd column and 1st row
    • | a12 a32 a42 |
    • | a13 a33 a43 |*((-1)^(2+1)) = V2
    • | a14 a34 a44 |
    Co-fector of 3rd column and 1st row
    • | a12 a22 a42 |
    • | a13 a23 a43 |*((-1)^(3+1)) = V3
    • | a14 a24 a44 |
    Co-fector of 4th column and 1st row
    • | a12 a22 a32 |
    • | a13 a23 a43 |*((-1)^(4+1)) = V4
    • | a14 a24 a34 |
    The value of the 4th order Det[A] = V1 + V2 + V3 + V4


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    AL 13 06. Value of 3rd order determinant

    Value of 2nd order determinant
    • | a11 a21 |
    • | a12 a22 | = a11*a22 - a21*a12
    Value of 3rd order determinant
    • | a11 a21 a31 |
    • | a12 a22 a32 |
    • | a13 a23 a33 |

    • = a11*a22*a33 +a21*a32*a13 +a31*a23*a12 -a31*a22*a13 -a21*a12*a33 -a11*a23*a32

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    AL 13 07. Answer

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    AL 13 08. Answer

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    AL 13 09. Answer


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    AL 13 10. Examples

    Example 1 : The given Det[A] = 0, find x
    • | 1 0 0 0 | = 0
    • | 0 1 0 0 |
    • | 0 0 x 2 |
    • | 0 0 8 x |
    Answer
    • | 1 0 0 0 |
    • | 0 1 0 0 |
    • | 0 0 x 2 |
    • | 0 0 8 x |
    • The co-factors of Det[A] are zero except the co-factor of 1st column and 1st raw
    • | 1 0 0 |
    • | 0 x 2 |*(-1)^(1+1)
    • | 0 8 x |
    • Similarly, we have
    • | x 2 | = x^2 - 16
    • | 8 x |
    • Since Det[A] = 0
    • Hence x^2 - 16 = 0
    • Hence x = 4 or x = -4

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    AL 13 11. Examples

    Example 1 : Find the value of x for the following determinant
      | 1 1 3 |
      | 2 2 1 | = 0
      | 3 x 2 |
    • Method 1
      • If x = 3, then column 1 and column 2 are same
      • If two columns are same then the value of determinant is zero
      • Hence x = 3
    • Method 2
      • 1*2*2 + 1*1*3 + 3*x*2 - 3*2*3 - 1*2*2 - 1*x*1 = 0
      • 4 + 3 + 6*x - 18 - 4 - x = 0
      • 5*x - 15 = 0
      • Hence x = 3
    Example 1 : Find the value of x for the following determinant
      | 2 x 1 |
      | 3 2 1 | = 0
      | 4 6 2 |
    • Method 1
      • Row 3 : Factor out 2 we have 2 3 1
      • If x = 3, then column 1 and column 2 are same
      • If two columns are same then the value of determinant is zero
      • Hence x = 3
    • Method 2
      • 2*2*2 + x*1*4 + 1*6*3 - 1*2*4 - x*3*2 - 2*6*1 = 0
      • 8 + 4*x + 18 - 8 - 6*x - 12 = 0
      • -2*x + 6 = 0
      • Hence x = 3

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    AL 13 00. Outline

    Properties of inverse matric
    • If Det[B] is the inverse matrix of Det[A]
    • Then Det[A]*Det[B] = I

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