2
Question
If A=⎡⎢⎣122212221⎤⎥⎦
and |(A2−pA−qI)|=0, then p+q=
If A=⎡⎢⎣122212221⎤⎥⎦
and |(A2−pA−qI)|=0, then p+q=
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Solution
The correct option is A 9
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∣∣
∣∣applying C1→C1+C2+C3|(A2−pA−qI)|=∣∣
∣∣25−5p−q8−2p8−2p25−5p−q9−p−q8−2p25−5p−q8−2p9−p−q∣∣
∣∣applying R2→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
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∣∣ ∣∣applying C1→C1+C2+C3|(A2−pA−qI)|=∣∣ ∣∣25−5p−q8−2p8−2p25−5p−q9−p−q8−2p25−5p−q8−2p9−p−q∣∣ ∣∣applying R2→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
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