행렬 대각합 성질

$\displaystyle A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} , B = \begin{pmatrix} p & q \\ r & s \end{pmatrix}$라 두고, 1행 1열의 성분과 2행 2열의 성분만 관찰하면 ㄱ. $\displaystyle A - B = \begin{pmatrix} a - p & \\ & d - s \end{pmatrix}$ $\therefore f \left( A - B \right) = ( a - p ) + ( d - s ) = ( a + d ) - ( p + s )$$= f ( A ) - f ( B )$ ㄴ. $\displaystyle AB = \begin{pmatrix} ap + br & \\ & cq + ds \end{pmatrix} , BA = \begin{pmatrix} ap + cq & \\ & br + ds \end{pmatrix}$ $\therefore f ( AB ) = ( ap + br ) + ( cq + ds ) = ( ap + cq ) + ( br + ds )$$= f ( BA )$ ㄷ. $\begin{aligned} f ( M ) = f \left( ACA ^{-1} - BCB ^{-1} \right) \\ = f \left( ACA ^{-1} \right) - f \left( BCB ^{-1} \right) \left( \because \text{ㄱ} \right) \\ = f \left( CA ^{-1} A \right) - f ( CB ^{-1} B ) \left( \because \text{ㄴ} \right) \\ = f \left( C \right) - f ( C ) = 0 \end{aligned}$ 그러므로 ㄱ, ㄴ, ㄷ 모두 옳다.