Question

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Answer 1

$\textsf{\large{\underline{Solution}:}}$

Given Information:

$\rm: \longmapsto\begin{bmatrix}\rm x&\rm 3x\\ \rm y&\rm 4y\end{bmatrix}\begin{bmatrix}\rm 2\\ \rm 1\end{bmatrix}=\begin{bmatrix}\rm5 \\ \rm12\end{bmatrix}$

Multiplying both the matrices, we get:

$\rm: \longmapsto\begin{bmatrix}\rm 2x + 3x\\ \rm 2y +4y \end{bmatrix}=\begin{bmatrix}\rm5 \\ \rm12\end{bmatrix}$

Therefore:

$\rm: \longmapsto\begin{bmatrix}\rm 5x\\ \rm 6y \end{bmatrix}=\begin{bmatrix}\rm5 \\ \rm12\end{bmatrix}$

Comparing both sides, we get:

$\rm: \longmapsto \begin{cases} \rm 5x = 5 \\ \rm 6y = 12 \end{cases}$

$\rm: \longmapsto (x, y) = (1,2)$

★ Which is our required answer.

$\textsf{\large{\underline{Verification}:}}$

Put x = 1 and y = 2, we get:

$\rm = \begin{bmatrix}\rm x&\rm 3x\\ \rm y&\rm 4y\end{bmatrix}\begin{bmatrix}\rm 2\\ \rm 1\end{bmatrix}$

$\rm = \begin{bmatrix}\rm 1&\rm 3\\ \rm 2&\rm 8\end{bmatrix}\begin{bmatrix}\rm 2\\ \rm 1\end{bmatrix}$

$\rm = \begin{bmatrix}\rm (1 \times 2) + (3 \times 1)\\ \rm (2 \times 2) + ( 8\times 1) \end{bmatrix}$

$\rm = \begin{bmatrix}\rm 2 + 3\\ \rm 4 + 8 \end{bmatrix}$

$\rm = \begin{bmatrix}\rm 5\\ \rm12 \end{bmatrix}$

Hence, our answer is correct (Verified).

$\textsf{\large{\underline{Learn More}:}}$

Matrix: A matrix is a rectangular arrangement of numbers in the form of horizontal and vertical lines.

Horizontal lines are called rows and vertical lines are called columns.

Order of Matrix: A matrix containing x rows and y column has order x × y and it has xy elements.

Different types of matrices:

Row Matrix: This type of matrices have only 1 row. Example:

$\rm:\longmapsto A=\begin{bmatrix}\rm 1&\rm 2&\rm 3\end{bmatrix}$

Column Matrix: This type of matrices have only 1 column. Example:

$\rm:\longmapsto A=\begin{bmatrix}\rm1\\ \rm2\\ \rm3\end{bmatrix}$

Square Matrix: In this type of matrix, number of rows and columns are equal. Example:

$\rm:\longmapsto A=\begin{bmatrix}\rm 1&\rm 2\\ \rm 3&\rm 4\end{bmatrix}$

Zero Matrix: It is a matrix with all elements present is zero. Example:

$\rm:\longmapsto A=\begin{bmatrix}\rm 0&\rm 0\\ \rm 0&\rm 0\end{bmatrix}$

Identity Matrix: In this type of matrix, diagonal element is 1 and remaining elements are zero. An Identity matrix is always a square matrix. Example:

$\rm:\longmapsto A=\begin{bmatrix}\rm 1&\rm 0\\ \rm 0&\rm 1\end{bmatrix}$

Answer 2

$\implies \tt \large{Solution}$

$\tt \fcolorbox{blue}{gy}{We \: have \: here}$

$\bf \huge \implies{[ \frac{x}{y} \: \: \frac{3x}{4y}]} \: \: [ \frac{2}{1} ] =[ \frac{5}{12} ]$

$\bf \huge\implies{[ \frac{2x + 3x}{2y + 4y} ] = [ \frac{5}{12} ]}$

$\bf \huge\implies{[ \frac{5x}{6y} ] = [ \frac{5}{12} ]}$

$\bf \large\implies{5x = 5 \: \: and \: \: 6y = 12}$

$\tt \huge \implies{x = 1 \: \: and \: \: y = 2}$

$\tt \huge \pink{hence}$

$\tt \huge \implies \fcolorbox{red}{white}{x = 1 \: \: and \: \: y = 2}$