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Question

If S=⎡⎢⎣011101110⎤⎥⎦ and A=⎡⎢⎣b+cc+ab−cc−bc+ba−bb−ca−ca+b⎤⎥⎦(a,b,c≠0), then SAS−1 is

A
symmetric matrix
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B
diagonal matrix
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C
invertible matrix
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D
singular matrix
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Solution

The correct options are A symmetric matrix B invertible matrix C diagonal matrixS=⎡⎢⎣011101110⎤⎥⎦,|S|=−1(−1)+1(1)=2∴S−1=AdjS|S|=12⎡⎢⎣−1111−1111−1⎤⎥⎦Now SA=⎡⎢⎣011101110⎤⎥⎦⎡⎢⎣b+cc+ab−cc−bc+aa−bb−ca−ca+b⎤⎥⎦=⎡⎢⎣02a2a2b02b2c2c0⎤⎥⎦SAS−1=⎡⎢⎣02a2a2b02b2c2c0⎤⎥⎦⋅12⎡⎢⎣−1111−1111−1⎤⎥⎦=⎡⎢⎣0aab0b00c⎤⎥⎦⎡⎢⎣−1111−1111−1⎤⎥⎦=⎡⎢⎣2a0002b0002c⎤⎥⎦∴SAS−1 is a diagonal matrix(SAS−1)T=⎡⎢⎣2a0002b0002c⎤⎥⎦=SAS−1 ∴ Symmetric∣∣SAS−1∣∣≠0∴SAS−1 is non-singular and invertible

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