$$ \left. \begin{array} { l l } { A = \left[ \begin{array} { l l } { 7 } & { 3 } \ { 6 } & { 1 } \end{array} ight] \quad B = \left[ \begin{array} { l l } { 4 } & { 2 } \ { 2 } & { 6 } \end{array} ight] } \ { A B = } & { B A = } \end{array} ight. $$

Question

$$ \left. \begin{array} { l l } { A = \left[ \begin{array} { l l } { 7 } & { 3 } \ { 6 } & { 1 } \end{array} ight] \quad B = \left[ \begin{array} { l l } { 4 } & { 2 } \ { 2 } & { 6 } \end{array} ight] } \ { A B = } & { B A = } \end{array} ight. $$

UpStudy AI · Accepted Answer

$$
A = \begin{bmatrix} 7 & 3 \ 6 & 1 \end{bmatrix}, \quad
B = \begin{bmatrix} 4 & 2 \ 2 & 6 \end{bmatrix}
$$

**Step 1. Compute $$AB$$:**

$$
AB = \begin{bmatrix}
7\cdot4 + 3\cdot2 & 7\cdot2 + 3\cdot6 \
6\cdot4 + 1\cdot2 & 6\cdot2 + 1\cdot6
\end{bmatrix}
$$

Calculate each entry:

- Top-left: $$7\cdot4 + 3\cdot2 = 28 + 6 = 34$$
- Top-right: $$7\cdot2 + 3\cdot6 = 14 + 18 = 32$$
- Bottom-left: $$6\cdot4 + 1\cdot2 = 24 + 2 = 26$$
- Bottom-right: $$6\cdot2 + 1\cdot6 = 12 + 6 = 18$$

Thus,

$$
AB = \begin{bmatrix} 34 & 32 \ 26 & 18 \end{bmatrix}
$$

**Step 2. Compute $$BA$$:**

$$
BA = \begin{bmatrix}
4\cdot7 + 2\cdot6 & 4\cdot3 + 2\cdot1 \
2\cdot7 + 6\cdot6 & 2\cdot3 + 6\cdot1
\end{bmatrix}
$$

Calculate each entry:

- Top-left: $$4\cdot7 + 2\cdot6 = 28 + 12 = 40$$
- Top-right: $$4\cdot3 + 2\cdot1 = 12 + 2 = 14$$
- Bottom-left: $$2\cdot7 + 6\cdot6 = 14 + 36 = 50$$
- Bottom-right: $$2\cdot3 + 6\cdot1 = 6 + 6 = 12$$

Thus,

$$
BA = \begin{bmatrix} 40 & 14 \ 50 & 12 \end{bmatrix}
$$
$$ AB = \begin{bmatrix} 34 & 32 \ 26 & 18 \end{bmatrix}, \quad BA = \begin{bmatrix} 40 & 14 \ 50 & 12 \end{bmatrix} $$

\( \left. \begin{array} { l l } { A = \left[ \begin{array} { l l } { 7 } & { 3 } \\ { 6 } & { 1 } \end{array} \right] \quad B = \left[ \begin{array} { l l } { 4 } & { 2 } \\ { 2 } & { 6 } \end{array} \right] } \\ { A B = } & { B A = } \end{array} \right. \)

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