Final answer:
To verify the positive definiteness of matrix S, we calculated the determinants of the upper-left submatrices (Δ1, Δ2, Δ3) and found that the ratios of these determinants should match the given pivots. We confirmed the positive definiteness by showing Δ1 = 2, Δ2 = 6, and Δ3 = 20 with the second and third pivot ratios being Δ2 / Δ1 and Δ3 / Δ2, respectively.
Step-by-step explanation:
The question is about verifying the positive definiteness of a matrix, S, by computing the upper left determinants and comparing their ratios to given pivots. To do this, we need to find the leading principal minors (determinants of the upper-left submatrices) of S. The matrix S is given by:
| 2 2 0 |
| 2 5 3 |
| 0 3 8 |
The first determinant (Δ1) is the top-left element, which is 2.
The second determinant (Δ2) is the determinant of the top-left 2x2 submatrix:
| 2 2 |
| 2 5 |
Calculating the determinant, Δ2 = (2)(5) - (2)(2) = 10 - 4 = 6.
The third determinant (Δ3) is the determinant of the entire matrix S:
| 2 2 0 |
| 2 5 3 |
| 0 3 8 |
Calculating the determinant using the rule of Sarrus or cofactor expansion, Δ3 = (2)(5)(8) + (2)(3)(0) + (0)(2)(3) - (0)(5)(0) - (2)(3)(2) - (2)(3)(8). Simplifying gives Δ3 = 80 - 12 - 48 = 20.
Now, we verify the ratios:
-
- Second pivot = Δ2 / Δ1 = 6 / 2 = 3
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- Third pivot = Δ3 / Δ2 = 20 / 6 ≈ 3.33
These are the ratios of the determinants and should match the given pivots to confirm that the matrix S is positive definite. The second and third pivots need to be provided to confirm the match.