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Chapter 5: Q28E (page 263)

TRUE OR FALSE

28. If Ais a symmetric matrix, vector v鈫� is in the image of A, and w鈫� is in the kernel of A, then the equation v鈫�路w鈫�=0 must hold.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Step-by-step solution Step 1: Definition of symmetric matrix

Inlinear algebra, a symmetric matrix is asquare matrixthat is equal to itstranspose. Formally, A is symmetric鈬�AAT

02

Check the given statement

For example,

The following3脳3 matrix is symmetric:

A=173745350

If w is the kernel of A then Aw鈫�=0

Therefore, rows of A are perpendicular to w鈫� as the dot product will be zero. As the matrix is symmetric, rows of A are equal to columns of A.

As vector v鈫� is in the image of A, then, v鈫�路w鈫�=0

Hence, the statement is true.

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