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Question

Let A=[XYZ] be a 3×3 orthogonal matrix with X,Y,Z as its column vectors. Then B=XXT+YYT

A
is a symmetric matrix
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B
is the 3×3 identity matrix
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C
satisfies B2=B
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D
satisfies BZ=O, O being the null matrix
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Solution

The correct option is D satisfies BZ=O, O being the null matrix
A is orthogonal matrix.
AAT=ATA=I
XTX=YTY=ZTZ=1
and XTY=XTZ=YTZ=YTX=ZTX=ZTY=0

B=XXT+YYT
BT=(XXT+YYT)T=XXT+YYT
Hence, B is symmetric.

B2=(XXT+YYT)(XXT+YYT)
=XXT(XXT+YYT)+YYT(XXT+YYT)
=XXT+YYT=B

BZ=(XXT+YYT)Z
=XXTZ+YYTZ
=O

We know that for any n×n identity matrix I and a n×1 non-zero vector A, IA=A
But here, BZ=O
Hence, B can't be the 3×3 identity matrix.

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