$$ \left. \begin{array} { l } { A = \left[ \begin{array} { c c } { - 2 } & { 3 } \ { 1 } & { 4 } \end{array} ight] \quad B = \left[ \begin{array} { c c c } { 5 } & { 6 } & { - 3 } \ { 4 } & { 1 } & { 2 } \end{array} ight] } \ { A B = } \ { B A = } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { A = \left[ \begin{array} { c c } { - 2 } & { 3 } \ { 1 } & { 4 } \end{array} ight] \quad B = \left[ \begin{array} { c c c } { 5 } & { 6 } & { - 3 } \ { 4 } & { 1 } & { 2 } \end{array} ight] } \ { A B = } \ { B A = } \end{array} ight. $$

UpStudy AI · Accepted Answer

We are given

$$
A = \begin{bmatrix} -2 & 3 \ 1 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 & -3 \ 4 & 1 & 2 \end{bmatrix}.
$$

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

Since $$A$$ is a $$2 \times 2$$ matrix and $$B$$ is a $$2 \times 3$$ matrix, the product $$AB$$ is defined and will be a $$2 \times 3$$ matrix.

Write $$AB = \begin{bmatrix} c_{11} & c_{12} & c_{13} \ c_{21} & c_{22} & c_{23} \end{bmatrix}$$, where each entry is computed by

$$
c_{ij} = \sum_{k=1}^{2} a_{ik} b_{kj}.
$$

1. For $$c_{11}$$:

$$
c_{11} = (-2)(5) + (3)(4) = -10 + 12 = 2.
$$

2. For $$c_{12}$$:

$$
c_{12} = (-2)(6) + (3)(1) = -12 + 3 = -9.
$$

3. For $$c_{13}$$:

$$
c_{13} = (-2)(-3) + (3)(2) = 6 + 6 = 12.
$$

4. For $$c_{21}$$:

$$
c_{21} = (1)(5) + (4)(4) = 5 + 16 = 21.
$$

5. For $$c_{22}$$:

$$
c_{22} = (1)(6) + (4)(1) = 6 + 4 = 10.
$$

6. For $$c_{23}$$:

$$
c_{23} = (1)(-3) + (4)(2) = -3 + 8 = 5.
$$

Thus,

$$
AB = \begin{bmatrix} 2 & -9 & 12 \ 21 & 10 & 5 \end{bmatrix}.
$$

---

**Step 2. Investigate $$BA$$:**

The matrix $$B$$ is $$2 \times 3$$ and $$A$$ is $$2 \times 2$$. For the product $$BA$$ to be defined, the number of columns of $$B$$ must equal the number of rows of $$A$$. Here, $$B$$ has 3 columns while $$A$$ has 2 rows, so

$$
\text{The product } BA \text{ is undefined}.
$$

---

**Final Answer:**

$$
AB = \begin{bmatrix} 2 & -9 & 12 \ 21 & 10 & 5 \end{bmatrix}, \quad BA \text{ is undefined.}
$$
$$ AB = \begin{bmatrix} 2 & -9 & 12 \ 21 & 10 & 5 \end{bmatrix} $$, and $$ BA $$ is undefined.

\( \left. \begin{array} { l } { A = \left[ \begin{array} { c c } { - 2 } & { 3 } \\ { 1 } & { 4 } \end{array} \right] \quad B = \left[ \begin{array} { c c c } { 5 } & { 6 } & { - 3 } \\ { 4 } & { 1 } & { 2 } \end{array} \right] } \\ { A B = } \\ { B A = } \end{array} \right. \)

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