Let (M ) and (N ) be two invertible square matrices over (R ) of order 2 such that (N ) is diagonal. Then (M N M⁻¹ ) is diagonal ____

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Correct answer: C. For all diagonal matrices (M )

Let the real matrices of order 2 (M= [ array ll a & b c & d array ] and N= [ array cc n₁ & 0 0 & n₂ array ] ) ( M ) and (N ) are invertible matrices, so (M⁻¹= 1 a d-b c [ array cc d & -b -c & a array ] ) Therefore, ( aligned M N M⁻¹ & = 1 a d-b c [ array ll a & b c & d array ] [ array cc n₁ & 0 0 & n₂ array ] [ array cc d & -b -c & a array ] & = 1 a d-b c [ array ll a & b c & d array ] [ array cc n₁ d & -n₁ b -n₂ c & n₂ a array ] & = 1 a d-b c [ array ll a n₁ d-b n₂ c & -a n₁ b+b n₂ a c n₁ d-d n₂ c & -c n₁ d+d n₂ a

Let (M ) and (N ) be two invertible square matrices over (R ) of order 2 such that (N ) is diagonal. Then (M N M⁻¹ ) is diagonal ____

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