Question

Find the value of \[x + y\]from the following equation:
\[2\left[ {\begin{array}{*{20}{c}}
  1&3 \\
  0&x
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
  y&0 \\
  1&2
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  5&6 \\
  1&8
\end{array}} \right]\].

Answer 1

Hint: To solve the question we have an idea of operations of matrices. At first we have to multiply by the matrix \[\left[ {\begin{array}{*{20}{c}}
  1&3 \\
  0&x
\end{array}} \right]\]by 2 and evaluate it and obtained \[2 \times 2\]matrix then added to the matrix\[\left[ {\begin{array}{*{20}{c}}
  y&0 \\
  1&2
\end{array}} \right]\]. Finally we have to equate the obtained matrix with\[\left[ {\begin{array}{*{20}{c}}
  5&6 \\
  1&8
\end{array}} \right]\]and find out the values of x and y which are then added to evaluate\[x + y\].

Complete step by step answer:
The matrix equation is given by
\[2\left[ {\begin{array}{*{20}{c}}
  1&3 \\
  0&x
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
  y&0 \\
  1&2
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  5&6 \\
  1&8
\end{array}} \right]\] …………………………….. (1)
We know that while a multiplying a matrix with a scalar, we must multiply each entry with the scalar for example when a matrix \[\left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right]\]is multiplied with a scalar ‘k’ then
\[k\left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {ka}&{kb} \\
  {kc}&{kd}
\end{array}} \right]\] ……………………………… (2)
Therefore applying this formula, we can have,
\[2\left[ {\begin{array}{*{20}{c}}
  1&3 \\
  0&x
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {2 \times 1}&{2 \times 3} \\
  {2 \times 0}&{2 \times x}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  2&6 \\
  0&{2x}
\end{array}} \right]\] ……………………………… (3)
Now substituting the value from eq. (3) in eq. (1), we will get
\[\left[ {\begin{array}{*{20}{c}}
  2&6 \\
  0&{2x}
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
  y&0 \\
  1&2
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  5&6 \\
  1&8
\end{array}} \right]\] ……………………………… (4)
We know that the addition is accomplished by adding corresponding elements. For example consider two matrices \[X = \left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right]\]and\[Y = \left[ {\begin{array}{*{20}{c}}
  e&f \\
  g&h
\end{array}} \right]\], then
\[X + Y = \left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
  e&f \\
  g&h
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {a + e}&{b + f} \\
  {c + g}&{d + h}
\end{array}} \right]\] ………………………. (5)
Applying this rule to LHS of eq. (4), we will get
\[\begin{gathered}
   \Rightarrow \left[ {\begin{array}{*{20}{c}}
  {2 + y}&{6 + 0} \\
  {0 + 1}&{2x + 2}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  5&6 \\
  1&8
\end{array}} \right] \\
   \Rightarrow \left[ {\begin{array}{*{20}{c}}
  {2 + y}&6 \\
  1&{2x + 2}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  5&6 \\
  1&8
\end{array}} \right] \\
\end{gathered} \]
………………………………. (6)
Now equating the corresponding elements of the matrices on both the sides of eq. (6), we will get,
\[\begin{gathered}
   \Rightarrow 2 + y = 5 \\
   \Rightarrow y = 3 \\
\end{gathered} \]
………………………………. (7)
\[\begin{gathered}
   \Rightarrow 2x + 2 = 8 \\
   \Rightarrow 2x = 6 \\
   \Rightarrow x = \dfrac{6}{2} = 3 \\
\end{gathered} \]
………………………………. (8)
Now adding the values of x and y from the equations (7) and (8), we will get,
\[x + y = 3 + 3 = 6\] ……………………………….. (9)
This is the required answer.

So, the correct answer is 6.

Note: Two matrices can be added if they have the same dimension that means same number of rows and columns. In addition to two matrices, the element of ith row and jth column in one matrix is added to the element of ith row and jth column in second matrix.