Question

multiply the matrices \\(a = \

$$\begin{bmatrix} 2 & -4 \\\\ 0 & 1 \\\\ 3 & 5 \\end{bmatrix}$$

\\) and \\(b = \

$$\begin{bmatrix} 1 & 6 \\\\ 0 & 7 \\end{bmatrix}$$

\\).

\\(\circ \

$$\begin{bmatrix} 2 & -16 \\\\ 0 & 7 \\\\ 3 & 53 \\end{bmatrix}$$

\\)
\\(\circ \

$$\begin{bmatrix} 2 & -16 & 0 \\\\ 13 & 3 & 53 \\end{bmatrix}$$

\\)

Explanation:

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<reasoning>

Check matrix dimensions

Using the Matrix Dimensions knowledge point
\[

$$\begin{aligned} &\text{Dimension of } A: 3 \times 2\\ &\text{Dimension of } B: 2 \times 2\\ &\text{Inner dimensions match: } 2 = 2\\ &\text{Resulting dimension of } AB: 3 \times 2 \end{aligned}$$

\]

Compute the product matrix elements

Using the Matrix Multiplication knowledge point
\[

$$\begin{aligned} C_{11} &= 2(1) + (-4)(0) = 2\\ C_{12} &= 2(6) + (-4)(7) = 12 - 28 = -16\\ C_{21} &= 0(1) + 1(0) = 0\\ C_{22} &= 0(6) + 1(7) = 7\\ C_{31} &= 3(1) + 5(0) = 3\\ C_{32} &= 3(6) + 5(7) = 18 + 35 = 53 \end{aligned}$$

\]

Construct the final product matrix

Using the Matrix Multiplication knowledge point
\[
AB =

$$\begin{bmatrix} 2 & -16 \\ 0 & 7 \\ 3 & 53 \end{bmatrix}$$

\]
</reasoning>

<answer>
<mcq-correct>\(

$$\begin{bmatrix} 2 & -16 \\ 0 & 7 \\ 3 & 53 \end{bmatrix}$$

\)</mcq-correct>
<mcq-option>\(

$$\begin{bmatrix} 2 & -16 & 0 \\ 13 & 3 & 53 \end{bmatrix}$$

\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Check matrix dimensions

Using the Matrix Dimensions knowledge point
\[

$$\begin{aligned} &\text{Dimension of } A: 3 \times 2\\ &\text{Dimension of } B: 2 \times 2\\ &\text{Inner dimensions match: } 2 = 2\\ &\text{Resulting dimension of } AB: 3 \times 2 \end{aligned}$$

\]

Compute the product matrix elements

Using the Matrix Multiplication knowledge point
\[

$$\begin{aligned} C_{11} &= 2(1) + (-4)(0) = 2\\ C_{12} &= 2(6) + (-4)(7) = 12 - 28 = -16\\ C_{21} &= 0(1) + 1(0) = 0\\ C_{22} &= 0(6) + 1(7) = 7\\ C_{31} &= 3(1) + 5(0) = 3\\ C_{32} &= 3(6) + 5(7) = 18 + 35 = 53 \end{aligned}$$

\]

Construct the final product matrix

Using the Matrix Multiplication knowledge point
\[
AB =

$$\begin{bmatrix} 2 & -16 \\ 0 & 7 \\ 3 & 53 \end{bmatrix}$$

\]
</reasoning>

<answer>
<mcq-correct>\(

$$\begin{bmatrix} 2 & -16 \\ 0 & 7 \\ 3 & 53 \end{bmatrix}$$

\)</mcq-correct>
<mcq-option>\(

$$\begin{bmatrix} 2 & -16 & 0 \\ 13 & 3 & 53 \end{bmatrix}$$

\)</mcq-option>
</answer>

<post_analysis>
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