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  • Fill in the blanks in each of the If A and B are symmetric matrices, then (i) AB - BA is a _________. (ii) BA - 2AB is a _________.

Fill in the blanks in each of the
If A and B are symmetric matrices, then
(i) AB - BA is a _________.
(ii) BA - 2AB is a _________.

Answers (1)

(i) AB - BA is a Skew Symmetric matrix

We are given that A’=A and B’=B

⇒ (AB-BA)’=(AB)’-(BA)’

⇒ (AB)’-(BA)’=B’A’-A’B’

⇒ B’A’-A’B’=BA-AB=-(AB-BA)

⇒ (AB-BA)’=-(AB-BA) (skew symmetric matrix)

\begin{aligned} &\text { For example, Let }\\ &A=\left[\begin{array}{ll} 1 & 3 \\ 3 & 2 \end{array}\right]\\ &B=\left[\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right]\\ &\Rightarrow \mathrm{AB}=\left[\begin{array}{ll} 7 & 5 \\ 7 & 8 \end{array}\right] \text { and } \mathrm{BA}=\left[\begin{array}{ll} 7 & 7 \\ 5 & 8 \end{array}\right]\\ &\Rightarrow A B-B A=\left[\begin{array}{cc} 0 & -2 \\ 2 & 0 \end{array}\right]\\ &\Rightarrow(A B-B A)^{\prime}=\left[\begin{array}{cc} 0 & 2 \\ -2 & 0 \end{array}\right]\\ &\Rightarrow=(A B-B A)=\left[\begin{array}{cc} 0 & 2 \\ -2 & 0 \end{array}\right] \end{aligned}

(ii) BA - 2AB is a Neither Symmetric nor Skew Symmetric matrix

Given A’=A and B’=B

⇒ (BA-2AB)’=(BA)’-(2AB)’

⇒ (BA)’-(2AB)’=A’B’-2B’A’

⇒ A’B’-2B’A’=AB-2BA=-(2BA-AB)

⇒ (BA-2AB)’=-(2BA-AB)

\begin{aligned} &\text { For example Let }\\ &A=\left[\begin{array}{ll} 1 & 3 \\ 3 & 2 \end{array}\right]\\ &B=\left[\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right]\\ &\Rightarrow \mathrm{AB}=\left[\begin{array}{ll} 7 & 5 \\ 7 & 8 \end{array}\right] \text { and } B A=\left[\begin{array}{ll} 7 & 7 \\ 5 & 8 \end{array}\right]\\ &\Rightarrow B A-2 A B=\left[\begin{array}{cc} 7 & -3 \\ -9 & 8 \end{array}\right] \end{aligned}

 

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