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In which pairs of matrices does [tex]$AB=BA$[/tex]?

[tex]$A=\left[\begin{array}{cc}1 & 0 \\ -2 & 1\end{array}\right], \quad B=\left[\begin{array}{cc}5 & 0 \\ 3 & 2\end{array}\right]$[/tex]

[tex]$A=\left[\begin{array}{cc}1 & 0 \\ -1 & 2\end{array}\right], \quad B=\left[\begin{array}{cc}3 & 0 \\ 6 & -3\end{array}\right]$[/tex]

[tex]$A=\left[\begin{array}{cc}1 & 0 \\ -1 & 2\end{array}\right], \quad B=\left[\begin{array}{cc}5 & 0 \\ 3 & 2\end{array}\right]$[/tex]

[tex]$A=\left[\begin{array}{cc}1 & 0 \\ -2 & 1\end{array}\right], \quad B=\left[\begin{array}{cc}3 & 0 \\ 6 & -3\end{array}\right]$[/tex]



Answer :

GinnyAnswer
To determine if the matrix pairs [tex]\(A\)[/tex] and [tex]\(B\)[/tex] commute, i.e., [tex]\(A B = B A\)[/tex], we must verify that the product of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] in both orders yields the same result. Here are the given matrix pairs:

1. [tex]\(A=\left[\begin{array}{cc}1 & 0 \\ -2 & 1\end{array}\right],\ B=\left[\begin{array}{ll}5 & 0 \\ 3 & 2\end{array}\right]\)[/tex]

2. [tex]\(A=\left[\begin{array}{cc}1 & 0 \\ -1 & 2\end{array}\right],\ B=\left[\begin{array}{cc}3 & 0 \\ 6 & -3\end{array}\right]\)[/tex]

3. [tex]\(A=\left[\begin{array}{cc}1 & 0 \\ -1 & 2\end{array}\right],\ B=\left[\begin{array}{ll}5 & 0 \\ 3 & 2\end{array}\right]\)[/tex]

4. [tex]\(A=\left[\begin{array}{cc}1 & 0 \\ -2 & 1\end{array}\right],\ B=\left[\begin{array}{cc}3 & 0 \\ 6 & -3\end{array}\right]\)[/tex]

Upon verifying which pairs satisfy [tex]\(A B = B A\)[/tex], we find the following:

1. For the pair
[tex]\[ A = \left[\begin{array}{cc}1 & 0 \\ -2 & 1\end{array}\right],\ B = \left[\begin{array}{ll}5 & 0 \\ 3 & 2\end{array}\right], \][/tex]
[tex]\(A B \neq B A\)[/tex].

2. For the pair
[tex]\[ A = \left[\begin{array}{cc}1 & 0 \\ -1 & 2\end{array}\right], B = \left[\begin{array}{cc}3 & 0 \\ 6 & -3\end{array}\right], \][/tex]
[tex]\(A B = B A\)[/tex].

3. For the pair
[tex]\[ A = \left[\begin{array}{cc}1 & 0 \\ -1 & 2\end{array}\right],\ B = \left[\begin{array}{ll}5 & 0 \\ 3 & 2\end{array}\right], \][/tex]
[tex]\(A B = B A\)[/tex].

4. For the pair
[tex]\[ A = \left[\begin{array}{cc}1 & 0 \\ -2 & 1\end{array}\right], B = \left[\begin{array}{cc}3 & 0 \\ 6 & -3\end{array}\right], \][/tex]
[tex]\(A B \neq B A\)[/tex].

Thus, the correct pairs where [tex]\(A B = B A\)[/tex] are the second and third pairs of matrices. The indices of these pairs are:
[tex]\[ [1, 2] \][/tex]

Therefore, the correct answers are:

- [tex]\( A=\left[\begin{array}{cc}1 & 0 \\ -1 & 2\end{array}\right], B=\left[\begin{array}{cc}3 & 0 \\ 6 & -3\end{array}\right] \)[/tex]
- [tex]\( A=\left[\begin{array}{cc}1 & 0 \\ -1 & 2\end{array}\right], B=\left[\begin{array}{ll}5 & 0 \\ 3 & 2\end{array}\right] \)[/tex]

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