Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Linear Algebra Bilinear Forms,Quadratic Forms
Let A = (ai, j) be a real symmetric 3 × 3 matrix. Consider the quadratic form Q(X1, X2, X3) = xt Ax where x = (X1, X2, X3)t.
Which of the following is true?
1
If Q(x1, x2, x3) is positive definite, then ai, j > 0 for all i ≠ j.
2
If Q(x1, x2, x3) is positive definite, then ai, i > 0 for all i.
3
If ai, j > 0 for all i ≠ j, then Q(x1, x2, x3) is positive definite.
4
If ai, i > 0 for all i, then Q(x1, x2, x3) is positive definite.
5
Question Not Attempted