Question

Find x and y, if \[x + y = \left[ {\begin{array}{*{20}{c}}
  7&0 \\
  2&5
\end{array}} \right]\] and \[x - y = \left[ {\begin{array}{*{20}{c}}
  3&0 \\
  0&3
\end{array}} \right]\].

Answer 1

Hint: A matrix is a rectangular array where numbers, symbols, or the variables are arranged in row and column. The numbers of the row and the column in the matrix tell the size or order of a matrix which means for a $2 \times 2$ matrix the number of rows and the column will be 2 each. \[M = 2 \times 2 = \left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right]\]
Matrix addition is the operation of the addition of each entry of a matrix with the corresponding elements of another matrix. If A and B are two matrices then their addition will be represented as (A+B).
Matrix subtraction is the operation of subtraction of elements of one matrix from the elements of another matrix represented as (A-B).

Complete step by step solution:
Both given matrix are $2 \times 2$ matrix hence addition and subtraction are possible
\[x + y = \left[ {\begin{array}{*{20}{c}}
  7&0 \\
  2&5
\end{array}} \right]\]- (i)
\[x - y = \left[ {\begin{array}{*{20}{c}}
  3&0 \\
  0&3
\end{array}} \right]\]- (ii)
Now let us find the value of \[x\]by adding both the equations (i) and (ii)
\[
  \left( {x + y} \right) + \left( {x - y} \right) = \left[ {\begin{array}{*{20}{c}}
  7&0 \\
  2&5
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
  3&0 \\
  0&3
\end{array}} \right] \\
  2x = \left[ {\begin{array}{*{20}{c}}
  {7 + 3}&{0 + 0} \\
  {2 + 0}&{5 + 3}
\end{array}} \right] \\
  2x = \left[ {\begin{array}{*{20}{c}}
  {10}&0 \\
  2&8
\end{array}} \right] \\
 \]
We can see \[x\]has a coefficient of 2 now we will cross multiply the equation and 2 will move to the denominator of the matrix, which will divide each element of the matrix
\[
  2x = \left[ {\begin{array}{*{20}{c}}
  {10}&0 \\
  2&8
\end{array}} \right] \\
  x = \dfrac{{\left[ {\begin{array}{*{20}{c}}
  {10}&0 \\
  2&8
\end{array}} \right]}}{2} \\
  x = \left[ {\begin{array}{*{20}{c}}
  {\dfrac{{10}}{2}}&{\dfrac{0}{2}} \\
  {\dfrac{2}{2}}&{\dfrac{8}{2}}
\end{array}} \right] \\
  x = \left[ {\begin{array}{*{20}{c}}
  5&0 \\
  1&4
\end{array}} \right] \\
 \]
Hence the value of \[x = \left[ {\begin{array}{*{20}{c}}
  5&0 \\
  1&4
\end{array}} \right]\], now for the value of y put the value of \[x\] in equation (i)
\[
  x + y = \left[ {\begin{array}{*{20}{c}}
  7&0 \\
  2&5
\end{array}} \right] \\
  y = \left[ {\begin{array}{*{20}{c}}
  7&0 \\
  2&5
\end{array}} \right] - x \\
  y = \left[ {\begin{array}{*{20}{c}}
  7&0 \\
  2&5
\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}
  5&0 \\
  1&4
\end{array}} \right] \\
  y = \left[ {\begin{array}{*{20}{c}}
  {7 - 5}&{0 - 0} \\
  {2 - 1}&{5 - 4}
\end{array}} \right] \\
  y = \left[ {\begin{array}{*{20}{c}}
  2&0 \\
  1&1
\end{array}} \right] \\
 \]
Hence the value of \[y = \left[ {\begin{array}{*{20}{c}}
  2&0 \\
  1&1
\end{array}} \right]\]


Note: Two matrices can be added or subtracted only when they have the same dimensions. You can add a $2 \times 3$ matrix with the other $2 \times 3$ matrix but you cannot add a $2 \times 3$ matrix with the $3 \times 2$ matrix. Matrices are useful while working with multiple linear equations.