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Chapter 8: Q16E (page 412)

16. Matrix [-2111-2111-2] is negative definite.

Short Answer

Expert verified

The given statement is FALSE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

For a matrix to be a negative definite, all the Eigenvalues of the matrix must be less than 0.

The quadratic form can be expanded as

qx1,x2,x3=-2x12-2x22-2x33+2x1x2+2x1x3+2x2x3

Re-write the above equation as follows:

qx1,x2,x3=-(x1-x2)2-(x1-x3)2-(x2-x3)2

Observe that q(1,1,1) = 0 which means that q is not a negative definite.

Since 0 is not less than zero the matrix cannot be negative definite. Thus, the given matrix is a non-negative definite.

02

Final Answer

The given statement is FALSE.

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