Mathematics Dictionary
Dr. K. G. Shih

    Figure 010 : Symmetrical matrix and Pascal triangle


  • Q01 | - Diagram : Symmetrical matrix of order 5
  • Q02 | - Symmetrical matrix of order 2
  • Q03 | - Symmetrical matrix of order 3
  • Q04 | - Symmetrical matrix of order 4

    Q01. Diagram

    Q1. What is the element of A^6 at 1st row and last column as shown in Figure
    • From Figure we see that the given answer is 126
    • The matrix is order 5 and we use sequence at r=4 (C4)
    • Power 6 is the 6th term of sequence at r=4 (C$)
    Q2. If the matrix is order 7, how to find A^6(1,6) ?
    • Use sequence at r=5 (C5)
    • Use 6th term of the sequence at r=5 (C5)
    • Use the given figure to find the answer

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    Q02. Symmetrical matrix of order 2

    1. Square following matrix : Find element at row 1 and column 2
    • | 1 1 |
    • | 0 1 |
    Answer
    • It is 2
    • It is n = 2 at column C1 in Pascal triangle
    2. Power 3 of following matrix : Find element at row 1 and column 2
    • | 1 1 |
    • | 0 1 |
    Answer
    • It is 3
    • It is n = 3 at column C1 in Pascal triangle
    3. Power 4 of following matrix : Find element at row 1 and column 2
    • | 1 1 |
    • | 0 1 |
    Answer
    • It is 4
    • It is n = 4 at column C1 in Pascal triangle

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    Q03. 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
    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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    4. 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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