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Mathematics Dictionary
Dr. K. G. Shih
Z041 : Matrix and Determinant


  • Q01 | - Detrminant of order 4
  • Q02 | - Two rows are identical
  • Q03 | - Two columns are identical
  • Q04 | - Symmetrical matrix of order 3
  • Q05 | - Symmetrical matrix of order 4
  • Q06 | - Reference


1. Determiant of order 4

Question : Find x
  • | 1 0 0 0 | = 0
  • | 0 1 0 0 |
  • | 0 0 x 2 |
  • | 0 0 8 x |
Solution
  • Co-factor of row 1 and column 1
    • | 1 0 0 |
    • | 0 x 2 | = 0
    • | 0 8 x |
  • Co-factor of row 1 and column 1 of determinant order 3
    • | x 2 |
    • | 8 x | = 0
  • Hence x^2 - 16 = 0
  • Hence x = 4 or x = - 4

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Q02. Two rows are identical

Question : Find values of the determinant
  • | 1 2 3 4 |
  • | 5 6 7 9 |
  • | 3 2 5 6 |
  • | 2 4 6 8 |
Answer
  • Take factor 2 from row 4
  • The row 1 and row 4 are identical
  • Using two identical row rule, we know the answer is zero

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Q03. Two columns are identical

Question : Find values of the determinant
  • | 1 2 3 2 |
  • | 4 6 7 8 |
  • | 3 2 5 6 |
  • | 2 4 6 4 |
Answer
  • Take factor 2 from column 4
  • The Column 1 and column 4 are identical
  • Using two identical column rule, we know the answer is zero

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Q04. Symmetrical matrix of order 3

1. Square following matrix : Find element at row 1 and column 2
  • | 1 1 1 |
  • | 0 1 1 |
  • | 0 0 1 |
Answer
  • It is C(3,1) = 3
  • It is n = 2 at column C2 in Pascal triangle
2. Power 3 of following matrix : Find element at row 1 and column 2
  • | 1 1 1 |
  • | 0 1 1 |
  • | 0 0 1 |
Answer
  • It is C(4,2) = 6
  • It is n = 3 at column C2 in Pascal triangle
3. Power 4 of following matrix : Find element at row 1 and column 2
  • | 1 1 1 |
  • | 0 1 1 |
  • | 0 0 1 |
Answer
  • It is C(5,3) = 10
  • It is n = 4 at column C2 in Pascal triangle
Example in Pascal triangle Summary : For order 3 symmetrical matrix
  • Power 2 of matrix order 3 = C(3,1) = 3
  • Power 3 of matrix order 3 = C(4,2) = 6
  • Power 4 of matrix order 3 = C(5,3) = 10
  • Power 5 of matrix order 3 = C(6,4) = 15
  • Etc.
  • This answer is the triangular number sequence

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5. Symmetrical matrix of order 4

Summary : For order 4 symmetrical matrix
  • Power 2 of matrix order 4 = C(4,1) = 4
  • Power 3 of matrix order 4 = C(5,2) = 10
  • Power 4 of matrix order 4 = C(6,3) = 20
  • Power 5 of matrix order 4 = C(7,4) = 35
  • Etc.
  • This is the triangular number sequence of 1st difference of sequence

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6. Refernce : Determinant

Reference
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