1
Question
If A=⎡⎢⎣102020213⎤⎥⎦, prove that A3−6A2+7A+2I=0
Open in App
Solution
A2=AA=⎡⎢⎣102020213⎤⎥⎦⎡⎢⎣102020213⎤⎥⎦
=⎡⎢⎣1+0+40+0+22+0+60+0+00+4+00+0+02+0+60+2+34+0+9⎤⎥⎦=⎡⎢⎣5280408513⎤⎥⎦
Now, A3=A2.A
=⎡⎢⎣5280408513⎤⎥⎦⎡⎢⎣102020213⎤⎥⎦
=⎡⎢⎣5+0+162+0+108+0+260+0+80+8+00+0+010+0+244+4+1516+0+39⎤⎥⎦
=⎡⎢⎣211234080342355⎤⎥⎦
∴A3−6A2+7A+2I
=⎡⎢⎣211234080342355⎤⎥⎦−6⎡⎢⎣5280408513⎤⎥⎦+7⎡⎢⎣102020213⎤⎥⎦+2⎡⎢⎣100010001⎤⎥⎦
=⎡⎢⎣211234080342355⎤⎥⎦−⎡⎢⎣3012480240483078⎤⎥⎦+⎡⎢⎣7014014014721⎤⎥⎦+⎡⎢⎣200020002⎤⎥⎦
=⎡⎢⎣21+7+212+0+034+14+00+0+08+14+20+0+034+14+023+7+055+21+2⎤⎥⎦−⎡⎢⎣3012480240483078⎤⎥⎦
=⎡⎢⎣3012480240483078⎤⎥⎦−⎡⎢⎣3012480240483078⎤⎥⎦
=⎡⎢⎣000000000⎤⎥⎦=0
∴A3−6A2+7A+2I=0
=⎡⎢⎣1+0+40+0+22+0+60+0+00+4+00+0+02+0+60+2+34+0+9⎤⎥⎦=⎡⎢⎣5280408513⎤⎥⎦
Now, A3=A2.A
=⎡⎢⎣5280408513⎤⎥⎦⎡⎢⎣102020213⎤⎥⎦
=⎡⎢⎣5+0+162+0+108+0+260+0+80+8+00+0+010+0+244+4+1516+0+39⎤⎥⎦
=⎡⎢⎣211234080342355⎤⎥⎦
∴A3−6A2+7A+2I
=⎡⎢⎣211234080342355⎤⎥⎦−6⎡⎢⎣5280408513⎤⎥⎦+7⎡⎢⎣102020213⎤⎥⎦+2⎡⎢⎣100010001⎤⎥⎦
=⎡⎢⎣211234080342355⎤⎥⎦−⎡⎢⎣3012480240483078⎤⎥⎦+⎡⎢⎣7014014014721⎤⎥⎦+⎡⎢⎣200020002⎤⎥⎦
=⎡⎢⎣21+7+212+0+034+14+00+0+08+14+20+0+034+14+023+7+055+21+2⎤⎥⎦−⎡⎢⎣3012480240483078⎤⎥⎦
=⎡⎢⎣3012480240483078⎤⎥⎦−⎡⎢⎣3012480240483078⎤⎥⎦
=⎡⎢⎣000000000⎤⎥⎦=0
∴A3−6A2+7A+2I=0
Suggest Corrections
0
Similar questions
View More
Join BYJU'S Learning Program
Explore more
Join BYJU'S Learning Program
