1
Question
Given A=⎡⎢⎣111241231⎤⎥⎦,B=[2334]
Find P such that BPA=[101010]
Find the absolute value of the sum of all the elements of P
Find P such that BPA=[101010]
Find the absolute value of the sum of all the elements of P
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Solution
Given BPA=[101010]
Pre-multiplying both sides by B−1
B−1BPA=B−1[101010]
⇒IPA=B−1[101010]
⇒PA=B−1[101010]....(1)
To find B−1
B=[2334]
∴|B|=∣∣∣2334∣∣∣=8−9=−1≠0
Let C be the matrix of cofactors of elements in |B|
C=[C11C12C21C22]
∴C11=4;C12=−3;C21=−3;C22=22
∴C=[4−3−32]
∴B−1=Adj.B|B|=C′−1=−C′=−[4−3−32]=[−433−2]
Now from (1)
PA=[−433−2]×[101010]
PA=[−43−43−23]
Post-multiplying both sides by A−1
PAA−1=[−43−43−23]A−1
⇒PI=[−43−43−23]A−1
⇒P=[−43−43−23]A−1...(2)
To find A−1
since, A=⎡⎢⎣111241231⎤⎥⎦
∴|A|=1(4−3)−1(2−2)+1(6−8)=1−0−2=−1≠0
Let C be the matrix of cofactors of elements in |A|
C=⎡⎢⎣C11C12C13C21C22C23C31C32C33⎤⎥⎦
=⎡⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣∣∣∣4131∣∣∣−∣∣∣2121∣∣∣∣∣∣2423∣∣∣−∣∣∣1131∣∣∣∣∣∣1121∣∣∣−∣∣∣1123∣∣∣∣∣∣1141∣∣∣−∣∣∣1121∣∣∣∣∣∣1124∣∣∣⎤⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥⎦
=⎡⎢⎣10−22−1−1−312⎤⎥⎦
C′=⎡⎢⎣12−30−11−2−12⎤⎥⎦
Adj A=⎡⎢⎣12−30−11−2−12⎤⎥⎦
A−1=Adj.A|A|=−Adj.A=⎡⎢⎣−1−2301−121−2⎤⎥⎦
From (2)
P=[−43−43−23]×⎡⎢⎣−1−2301−121−2⎤⎥⎦=[4+0−88+3−4−12−3+8−3−0+6−6−2+39+2−6]
∴P=[−47−73−55]
Pre-multiplying both sides by B−1
B−1BPA=B−1[101010]
⇒IPA=B−1[101010]
⇒PA=B−1[101010]....(1)
To find B−1
B=[2334]
∴|B|=∣∣∣2334∣∣∣=8−9=−1≠0
Let C be the matrix of cofactors of elements in |B|
C=[C11C12C21C22]
∴C11=4;C12=−3;C21=−3;C22=22
∴C=[4−3−32]
∴B−1=Adj.B|B|=C′−1=−C′=−[4−3−32]=[−433−2]
Now from (1)
PA=[−433−2]×[101010]
PA=[−43−43−23]
Post-multiplying both sides by A−1
PAA−1=[−43−43−23]A−1
⇒PI=[−43−43−23]A−1
⇒P=[−43−43−23]A−1...(2)
To find A−1
since, A=⎡⎢⎣111241231⎤⎥⎦
∴|A|=1(4−3)−1(2−2)+1(6−8)=1−0−2=−1≠0
Let C be the matrix of cofactors of elements in |A|
C=⎡⎢⎣C11C12C13C21C22C23C31C32C33⎤⎥⎦
=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣∣∣∣4131∣∣∣−∣∣∣2121∣∣∣∣∣∣2423∣∣∣−∣∣∣1131∣∣∣∣∣∣1121∣∣∣−∣∣∣1123∣∣∣∣∣∣1141∣∣∣−∣∣∣1121∣∣∣∣∣∣1124∣∣∣⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦
=⎡⎢⎣10−22−1−1−312⎤⎥⎦
C′=⎡⎢⎣12−30−11−2−12⎤⎥⎦
Adj A=⎡⎢⎣12−30−11−2−12⎤⎥⎦
A−1=Adj.A|A|=−Adj.A=⎡⎢⎣−1−2301−121−2⎤⎥⎦
From (2)
P=[−43−43−23]×⎡⎢⎣−1−2301−121−2⎤⎥⎦=[4+0−88+3−4−12−3+8−3−0+6−6−2+39+2−6]
∴P=[−47−73−55]
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