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Let-A-and-B-is-3-3-matrix-of-equal-number-where-A-symmetric-matrix-B-skew-symmetric-matrix-and-the-relation-A-B-A-B-A-B-A-B-then-the-value-of-k-AB-T-1-k-AB-a-1-




Question Number 19589 by gourav~ last updated on 13/Aug/17
Let A and B is 3×3 matrix of equal number  where A=symmetric matrix   ....B=skew symmetric matrix  and the relation... (A+B)(A−B)=(A−B)(A+B)  then..the value of.. ... k      (AB)^T =(−1)^k (AB)  (a) −1                    (c) 2    (b) 1                          (d) 3
$${Let}\:{A}\:{and}\:{B}\:{is}\:\mathrm{3}×\mathrm{3}\:{matrix}\:{of}\:{equal}\:{number} \\ $$$${where}\:{A}={symmetric}\:{matrix}\: \\ $$$$….{B}={skew}\:{symmetric}\:{matrix} \\ $$$${and}\:{the}\:{relation}…\:\left({A}+{B}\right)\left({A}−{B}\right)=\left({A}−{B}\right)\left({A}+{B}\right) \\ $$$${then}..{the}\:{value}\:{of}..\:…\:{k} \\ $$$$\:\:\:\:\left({AB}\right)^{{T}} =\left(−\mathrm{1}\right)^{{k}} \left({AB}\right) \\ $$$$\left({a}\right)\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({c}\right)\:\mathrm{2} \\ $$$$ \\ $$$$\left({b}\right)\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({d}\right)\:\mathrm{3} \\ $$
Answered by dioph last updated on 15/Aug/17
A^2 −AB+BA−B^2 =A^2 +AB−BA+B^2   2AB = 2BA  AB = BA  (AB)^T  = (BA)^T   (AB)^T  = A^T B^T   (AB)^T  = A(−B)  (AB)^T  = (−1)(AB)
$${A}^{\mathrm{2}} −{AB}+{BA}−{B}^{\mathrm{2}} ={A}^{\mathrm{2}} +{AB}−{BA}+{B}^{\mathrm{2}} \\ $$$$\mathrm{2}{AB}\:=\:\mathrm{2}{BA} \\ $$$${AB}\:=\:{BA} \\ $$$$\left({AB}\right)^{{T}} \:=\:\left({BA}\right)^{{T}} \\ $$$$\left({AB}\right)^{{T}} \:=\:{A}^{{T}} {B}^{{T}} \\ $$$$\left({AB}\right)^{{T}} \:=\:{A}\left(−{B}\right) \\ $$$$\left({AB}\right)^{{T}} \:=\:\left(−\mathrm{1}\right)\left({AB}\right) \\ $$

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