Question

If \[\begin{array}{*{20}{c}}{\left[ {\begin{array}{*{20}{c}}{x + y}&{2x + z}\\{x - y}&{2z + w}\end{array}} \right]}& = &{\left[ {\begin{array}{*{20}{c}}4&7\\0&{10}\end{array}} \right]}\end{array}\] then the value of \[x\], \[y\],\[z\]and \[w\]are,
A. \[2,2,3,4\]
B. \[2,3,1,2\]
C. \[3,3,0,1\]
D. None of these

Answer 1

Hint: In the question, we have given the \[2 \times 2\] matrix. We will compare each of the elements of the given matrix. After that, we will solve the equations and determine the value of the \[x\] and \[y\] then we will substitute the value of the \[x\] and \[y\] in another equation to determine the value of the \[z\] and \[w\] hence, we will get our desired answer.

Complete step by step Solution:
In the question, we have given the matrix such as

\[\begin{array}{*{20}{c}}{ \Rightarrow \left[ {\begin{array}{*{20}{c}}{x + y}&{2x + z}\\{x - y}&{2z + w}\end{array}} \right]}& = &{\left[ {\begin{array}{*{20}{c}}4&7\\0&{10}\end{array}} \right]}\end{array}\]
From the above matrix, we will compare each of the elements of the given matrix. Therefore, we will get four equations such as
\[ \Rightarrow \begin{array}{*{20}{c}}{x + y}& = &4\end{array}\] ---------- (1)
\[ \Rightarrow \begin{array}{*{20}{c}}{x - y}& = &0\end{array}\]---------- (2)
And
\[ \Rightarrow \begin{array}{*{20}{c}}{2x + z}& = &7\end{array}\]--------- (3)
\[ \Rightarrow \begin{array}{*{20}{c}}{2z + w}& = &{10}\end{array}\]-------- (4)
Now, we will solve equation (1) and (2) to determine the value of the \[x\]and \[y\]. By adding the equation (1) and (2), we will get
\[ \Rightarrow \begin{array}{*{20}{c}}{\left( {x + y} \right) + \left( {x - y} \right)}& = &{4 + 0}\end{array}\]
\[ \Rightarrow \begin{array}{*{20}{c}}{2x}& = &4\end{array}\]
\[ \Rightarrow \begin{array}{*{20}{c}}x& = &2\end{array}\]
Now we will put the value of the \[x\]in the equation (1). Therefore, we will get
\[ \Rightarrow \begin{array}{*{20}{c}}{2 + y}& = &4\end{array}\]
\[ \Rightarrow \begin{array}{*{20}{c}}y& = &2\end{array}\]
Now we will put the value of \[x\] in equation (3). Therefore, we will get
\[ \Rightarrow \begin{array}{*{20}{c}}{2 \times 2 + z}& = &7\end{array}\]
\[ \Rightarrow \begin{array}{*{20}{c}}z& = &3\end{array}\]
And then we will put the value of the \[z\]in equation (4). Therefore, we will get
\[ \Rightarrow \begin{array}{*{20}{c}}{2 \times 3 + w}& = &{10}\end{array}\]
\[ \Rightarrow \begin{array}{*{20}{c}}w& = &4\end{array}\]
Finally, we have got the desired answer.

Therefore, the correct option is (A).

Note: In this question, the first point is to keep in mind that elements of the matrix can be compared with elements of another matrix. And then we will solve the equations to get the desired answer. In these types of problems, students are likely to make mistakes because it includes a number of formulas to remember. It is also important to keep in mind that signs are important in these types of problems as wrong signs may lead to wrong solutions.