Answered

Connect with a global community of experts on IDNLearn.com. Ask any question and receive timely, accurate responses from our dedicated community of experts.

Select all the correct answers.

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]

Sagot :

GinnyAnsweridn
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]
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.