Question: Write the following quadratic forms in matrix notation and determine if they are positive definite: (a) x2 + 4xz + 2y2 + 8yz + 12z2

Write the following quadratic forms in matrix notation and determine if they are positive definite:
(a) x2 + 4xz + 2y2 + 8yz + 12z2
(b) 3x2 - 2xy2 - 8xy + xz + z2
(c) x2 + 2xy + 2y2 -4xz - 6yz + 6z2
(d) 3x2 - x22 + 5x23 + 4x|x2 - 7x1x3 + 9x2x3
(e) x21 + 4x1x2 - 2x1x3 + 5x22 - 2x2x4 + 6x23 - x3x4 +4x2i

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