Show that if A and B are square matrices such that #AB = BA# then: #(A+B)^2=A^2+2AB+B^2#?

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Answer 1 · 2018-03-04T10:19:34

Kindly refer to the Explanation.

Since #A and B# are square matrices, all the multiplications

reqd. in the Question are defined.

Now, #(A+B)^2=(color(red)A+color(red)B)*(color(green)A+color(green)B)#,

#=color(red)A(color(green)(A+B))+color(red)B(color(green)(A+B))#,

#=A*A+A*B+B*A+B*B#,

#=A^2+A*B+A*B+B^2.......[because, A*B=B*A," Given]"#,

#=A^2+2A*B+B^2, i.e., #

#=A^2+2AB+B^2#, as desired!