Question

If \[A = \left( {\begin{array}{*{20}{c}}
  2&1&{ - 8} \\
  { - 9}&4&7 \\
  { - 1}&9&3 \\
  1&4&1
\end{array}} \right)\], \[B = \left( {\begin{array}{*{20}{c}}
  3&3&1 \\
  3&{ - 3}&{ - 3} \\
  7&{ - 2}&{ - 4} \\
  { - 9}&1&{ - 7}
\end{array}} \right)\] and \[2A - B = \left( {\begin{array}{*{20}{c}}
   \cdot & \cdot &a \\
   \cdot & \cdot & \cdot \\
   \cdot & \cdot & \cdot \\
   \cdot & \cdot & \cdot
\end{array}} \right)\] what is the value of element a?

Answer 1

Hint: Here in this question, we have to find the value of element a in the resultant matrix. The resultant matrix 2A-B can be find by firstly multiply A matrix with 2 and subtract matrix B from it by using the basic algebraic operation like addition, subtraction and multiplication in to the matrix a we get the resultant matrix in that first row third column element is the required value of a.

Complete step by step solution:
A matrix is a collection of numbers arranged into a fixed number of rows and columns, usually the numbers are real numbers. Generally matrices are represented as \[A = {\left[ a \right]_{ij}}\] where i represents the number of rows and j represents the number of columns, \[1 \leqslant i,j \leqslant n\]
Now consider the given matrix A and B
\[A = \left( {\begin{array}{*{20}{c}}
  2&1&{ - 8} \\
  { - 9}&4&7 \\
  { - 1}&9&3 \\
  1&4&1
\end{array}} \right)\] and \[B = \left( {\begin{array}{*{20}{c}}
  3&3&1 \\
  3&{ - 3}&{ - 3} \\
  7&{ - 2}&{ - 4} \\
  { - 9}&1&{ - 7}
\end{array}} \right)\]
Here, in this both matrices A and B having 4 rows and 3 columns the resultant must also have 4 rows and 3 columns
Now, consider
\[ \Rightarrow \,\,2A - B\]
Substitute A and B matrix in it, then
\[ \Rightarrow \,\,2\left( {\begin{array}{*{20}{c}}
  2&1&{ - 8} \\
  { - 9}&4&7 \\
  { - 1}&9&3 \\
  1&4&1
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
  3&3&1 \\
  3&{ - 3}&{ - 3} \\
  7&{ - 2}&{ - 4} \\
  { - 9}&1&{ - 7}
\end{array}} \right)\]
Multiply 2 to every single value (element) in the whole matrix A., then
\[ \Rightarrow \,\,\left( {\begin{array}{*{20}{c}}
  4&2&{ - 16} \\
  { - 18}&8&{14} \\
  { - 2}&{18}&6 \\
  2&8&2
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
  3&3&1 \\
  3&{ - 3}&{ - 3} \\
  7&{ - 2}&{ - 4} \\
  { - 9}&1&{ - 7}
\end{array}} \right)\]
Now, subtract the respective elements in matrix B with the elements in matrix A, then we get
\[ \Rightarrow \,\,\left( {\begin{array}{*{20}{c}}
  {4 - 3}&{2 - 3}&{ - 16 - 1} \\
  { - 18 - 3}&{8 - \left( { - 3} \right)}&{14 - \left( { - 3} \right)} \\
  { - 2 - 7}&{18 - \left( { - 2} \right)}&{6 - \left( { - 4} \right)} \\
  {2 - \left( { - 9} \right)}&{8 - 1}&{2 - \left( { - 7} \right)}
\end{array}} \right)\]
\[ \Rightarrow \,\,\left( {\begin{array}{*{20}{c}}
  1&{ - 1}&{ - 17} \\
  { - 21}&{11}&{17} \\
  { - 9}&{20}&{10} \\
  {11}&7&9
\end{array}} \right)\]
Now, consider
\[ \Rightarrow \,\,2A - B = \left( {\begin{array}{*{20}{c}}
   \cdot & \cdot &a \\
   \cdot & \cdot & \cdot \\
   \cdot & \cdot & \cdot \\
   \cdot & \cdot & \cdot
\end{array}} \right)\]
Substitute the \[2A - B\] matrix
\[ \Rightarrow \,\,\left( {\begin{array}{*{20}{c}}
  1&{ - 1}&{ - 17} \\
  { - 21}&{11}&{17} \\
  { - 9}&{20}&{10} \\
  {11}&7&9
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
   \cdot & \cdot &a \\
   \cdot & \cdot & \cdot \\
   \cdot & \cdot & \cdot \\
   \cdot & \cdot & \cdot
\end{array}} \right)\]
By comparing the two matrix
The value of a is
\[ \Rightarrow \,\,a = - 17\]
Hence, the value is -17.
So, the correct answer is “a = - 17”.

Note: The arithmetic operations are applied for the matrix also. While multiplying the terms we must know the table of multiplications, if we go wrong in the table of multiplication the whole answer will go wrong. By doing the operation which is mentioned in the question and then we determine the value