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Question

Suppose A is a 3×3 matrix. If for every column vector X, XTAX=0 and a23=1008 , then the sum of the digits of a32 is:


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Solution

Use the properties of transpose of a matrix

Given, XTAX=0 and a23=-1008

Let A be any 3×3 square matrix, such that

A=a11a12a13a21a22a23a31a32a33

Then

XTAX=(XTAX)T=XTATX=-XTAX

XTAX=0

Therefore, XTATX=-XTAX

AT=-A

a23=-a32

or

a32=-a23

a32=(-1008)=1008

Therefore, the sum of digits=1+0+0+8=9

Hence, the sum of digits of a32 is 9


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