Question #150834
if A= [ 1 -3 2 and B= [ 2 1 5
4 1 -1 -1 -2 -2
-3 2 5 ] 3 1 2 ]

then find (AB)^-1
1
Expert's answer
2020-12-20T18:54:18-0500

Given matrices are

A=[132411325]A = \begin{bmatrix} 1 & -3 & 2 \\ 4 & 1 & -1 \\ 3 & 2 & 5 \end{bmatrix} and B=[215122312]B = \begin{bmatrix} 2 & 1 & 5 \\ -1 & -2 & -2 \\ 3 & 1 & 2 \end{bmatrix}



Now, AB=[132411325][215122312]=[11915411619421]AB = \begin{bmatrix} 1 & -3 & 2 \\ 4 & 1 & -1 \\ 3 & 2 & 5 \end{bmatrix} \begin{bmatrix} 2 & 1 & 5 \\ -1 & -2 & -2 \\ 3 & 1 & 2 \end{bmatrix} = \begin{bmatrix} 11 & 9 & 15 \\ 4 & 1 & 16 \\ 19 & 4 & 21 \end{bmatrix}


Determinant of the matrix AB is 1471.

Co-factors are:

a11=(2164)=43a_{11} = (21-64) = -43

a12=(84304)=220a_{12}=-(84-304)=220

a13=(1619)=3a_{13}= (16-19) = -3

a21=(18960)=129a_{21}=-(189-60)=-129

a22=(231285)=54a_{22}=(231-285)=-54

a23=(44171)=127a_{23}=-(44-171)=127

a31=(14415)=129a_{31}=(144-15)=129

a32=(17660)=116a_{32}=-(176-60)=-116

a33=(1136)=25a_{33}=(11-36)=-25



adj(AB)=[4312912922054116312725]adj (AB) = \begin{bmatrix} -43 & -129 & 129 \\ 220 & -54 & -116 \\ -3 & 127 & -25 \end{bmatrix}

adj(AB)=adj (AB) =

Then inverse of the matrix AB is,

(AB)1=11471[4312912922054116312725](AB)^{-1} = \frac{1}{1471}\begin{bmatrix} -43 & -129 & 129 \\ 220 & -54 & -116 \\ -3 & 127 & -25 \end{bmatrix}




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