Question

How do you solve \[2x+A=B\] given \[A=\left[ \begin{matrix}
   2 \\
   9 \\
   -2 \\
\end{matrix}\begin{matrix}
   -8 \\
   5 \\
   3 \\
\end{matrix} \right]\] and \[B=\left[ \begin{matrix}
   -6 \\
   1 \\
   8 \\
\end{matrix}\begin{matrix}
   2 \\
   -5 \\
   5 \\
\end{matrix} \right]\]?

Answer 1

Hint: Consider x as a variable matrix and solve for the value of x. Take the matrix A to the R.H.S. and subtract the elements of matrix A from the corresponding elements of matrix B. Now, divide both the sides with 2 to make the coefficient of x equal to 1. Accordingly divide each element of the resultant matrix in the R.H.S. by 2 and get the answer.

Complete step by step solution:
Here, we have been provided with the equation \[2x+A=B\] and the values of A and B are given in matrix form. We are asked to solve for the value of x, that means we have to determine the matrix x.
Now, leaving the variable x in the L.H.S. and taking the matrices A and B to the R.H.S., we get,
\[\Rightarrow 2x=B-A\]
Substituting the given matrix form of A and B, we get,
\[\Rightarrow 2x=\left[ \begin{matrix}
   -6 \\
   1 \\
   8 \\
\end{matrix}\begin{matrix}
   2 \\
   -5 \\
   5 \\
\end{matrix} \right]-\left[ \begin{matrix}
   2 \\
   9 \\
   -2 \\
\end{matrix}\begin{matrix}
   -8 \\
   5 \\
   3 \\
\end{matrix} \right]\]
Here, we need to perform the subtraction operation on two matrices. So, according to the subtraction property of matrices we need to subtract each element of matrix A from the corresponding elements of matrix B. So, we get,
\[\begin{align}
  & \Rightarrow 2x=\left[ \begin{matrix}
   -6-2 \\
   1-9 \\
   8-\left( -2 \right) \\
\end{matrix}\begin{matrix}
   2-\left( -8 \right) \\
   -5-5 \\
   5-3 \\
\end{matrix} \right] \\
 & \Rightarrow 2x=\left[ \begin{matrix}
   -8 \\
   -8 \\
   10 \\
\end{matrix}\begin{matrix}
   10 \\
   -10 \\
   2 \\
\end{matrix} \right] \\
\end{align}\]
Dividing both the sides with 2, we get,
\[\Rightarrow x=\dfrac{1}{2}\left[ \begin{matrix}
   -8 \\
   -8 \\
   10 \\
\end{matrix}\begin{matrix}
   10 \\
   -10 \\
   2 \\
\end{matrix} \right]\]
Now, the division or multiplication property in a matrix states that when we multiply or divide a matrix with a scalar then each element of that matrix is to be divided or multiplied with scalar. So, we get,
\[\Rightarrow x=\left[ \begin{matrix}
   -4 \\
   -4 \\
   5 \\
\end{matrix}\begin{matrix}
   5 \\
   -5 \\
   1 \\
\end{matrix} \right]\]
Hence, the above matrix obtained is our answer.

Note: One may note that here you must not consider the variable x as a simple linear variable. Actually, it is a matrix having six variables which can be represented in matrix form as \[x=\left[ \begin{matrix}
   {{x}_{1}} \\
   {{x}_{3}} \\
   {{x}_{5}} \\
\end{matrix}\begin{matrix}
   {{x}_{2}} \\
   {{x}_{4}} \\
   {{x}_{6}} \\
\end{matrix} \right]\]. Remember that we can add or subtract two or more matrices only when they have the same order. In the above question the matrices were of the order \[\left( 3\times 2 \right)\] where ‘3’ represents the number of rows and ‘2’ represents the number of columns. You must remember the division and multiplication property of a scalar with a given matrix, otherwise you will get confused in the last step of the solution.