Question

411 12 a13 [b1 b21 C11 C12 C13 31 32 33 b13 b23 C31 C32 C33 Theory Prove the following (Assume A-1 and C both exist.) 1. (A-B)' = B' . A, 2. (A C) C.A-.

Answer 1

$1.~(A.B)^\prime=\left(\begin{matrix} a_{11}b_{11} +a_{12}b_{12}+a_{13}b_{13}& a_{21}b_{11} +a_{22}b_{12}+a_{23}b_{13}&a_{31}b_{11} +a_{32}b_{12}+a_{33}b_{13}\\ a_{11}b_{21} +a_{12}b_{22}+a_{13}b_{23}&a_{21}b_{21} +a_{22}b_{22}+a_{23}b_{23}&a_{31}b_{21} +a_{32}b_{22}+a_{33}b_{23}& \\ \end{matrix} \right )\\ B^\prime A^\prime=\begin{pmatrix} b_{11} & b_{12} &b_{13} \\ b_{21}&b_{22} &b_{23} \end{pmatrix}\begin{pmatrix} a_{11} & a_{21} &a_{31} \\ a_{12}& a_{22} &a_{23} \\ a_{31} &a_{32} &a_{33} \end{pmatrix}=\left(\begin{matrix} a_{11}b_{11} +a_{12}b_{12}+a_{13}b_{13}& a_{21}b_{11} +a_{22}b_{12}+a_{23}b_{13}&a_{31}b_{11} +a_{32}b_{12}+a_{33}b_{13}\\ a_{11}b_{21} +a_{12}b_{22}+a_{13}b_{23}&a_{21}b_{21} +a_{22}b_{22}+a_{23}b_{23}&a_{31}b_{21} +a_{32}b_{22}+a_{33}b_{23}& \\ \end{matrix} \right )\\ Hence~(AB)^\prime=B^\prime A^\prime$$2.~[C^{-1}A^{-1}][AC]=C^{-1}[A^{-1}A]C=C^{-1}C=I_3=Identity ~matrix~of~order~3.\\ or,~(A.C)^{-1}=C^{-1}A^{-1}.\\ 3.~(A^{-1})^\prime A^\prime=(AA^{-1})^\prime=I_3^\prime=I_3\\ or,~(A^{-1})^\prime=(A^\prime)^{-1}$