If A=⎡⎢⎣122212221⎤⎥⎦ and |(A2−pA−qI)|=0, then p+q=
A
9
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B
8
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C
7
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D
10
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Solution
The correct option is A9 A=⎡⎢⎣122212221⎤⎥⎦⇒A2=⎡⎢⎣988898889⎤⎥⎦⇒A2−pA−qI=⎡⎢⎣9−p−q8−2p8−2p8−2p9−p−q8−2p8−2p8−2p9−p−q⎤⎥⎦|(A2−pA−qI)|=∣∣
∣∣9−p−q8−2p8−2p8−2p9−p−q8−2p8−2p8−2p9−p−q∣∣
∣∣applyingC1→C1+C2+C3|(A2−pA−qI)|=∣∣
∣∣25−5p−q8−2p8−2p25−5p−q9−p−q8−2p25−5p−q8−2p9−p−q∣∣
∣∣applyingR2→R2−R1R3→R3−R1=∣∣
∣∣25−5p−q8−2p8−2p01+p−q0001+p−q∣∣
∣∣0=(25−5p−q)(1+p−q)2q−p=1 and 5p+q=25⇒p=4;q=5