the matrix $r = \\begin{bmatrix}12 \\\\ 4\\end{bmatrix}$ represents the length of 12 in. and the width of 4…

the matrix $r = \\begin{bmatrix}12 \\\\ 4\\end{bmatrix}$ represents the length of 12 in. and the width of 4 in. of a rectangle. if matrix multiplication is used to double the length and width of the rectangle, which rectangle represents the result of the matrix multiplication?

the matrix $r = \\begin{bmatrix}12 \\\\ 4\\end{bmatrix}$ represents the length of 12 in. and the width of 4 in. of a rectangle. if matrix multiplication is used to double the length and width of the rectangle, which rectangle represents the result of the matrix multiplication?

Answer

Explanation:

Step1: Define the scalar matrix

To double each element of the matrix $R=\begin{bmatrix}12\4\end{bmatrix}$, we multiply it by the scalar matrix $\begin{bmatrix}2&0\0&2\end{bmatrix}$ (in general for 2 - D scaling). But since we are dealing with a column - vector representing length and width, we can simply multiply by the scalar 2.

Step2: Perform matrix - scalar multiplication

We multiply the scalar 2 by the matrix $R$. The multiplication is done element - wise. So, $2\times\begin{bmatrix}12\4\end{bmatrix}=\begin{bmatrix}2\times12\2\times4\end{bmatrix}=\begin{bmatrix}24\8\end{bmatrix}$.

Answer:

The rectangle with length 24 in. and width 8 in.