Question

What is the result when you add the matrix \[\left[ \begin{matrix}
   4 & 5 \\
\end{matrix} \right]\] to the matrix \[\left[ \begin{matrix}
   7 & -3 \\
\end{matrix} \right]\] and multiply the result by 2?
(a) \[\left[ \begin{matrix}
   2 & 6 \\
\end{matrix} \right]\]
(b) \[\left[ \begin{matrix}
   11 & 2 \\
\end{matrix} \right]\]
(c) \[\left[ \begin{matrix}
   22 & 4 \\
\end{matrix} \right]\]
(d) \[\left[ \begin{matrix}
   28 & -15 \\
\end{matrix} \right]\]
(e) \[\left[ \begin{matrix}
   54 & -30 \\
\end{matrix} \right]\]

Answer 1

Hint: Assume the given matrices as A and B and perform the addition operation by using the addition property in matrices. Add the \[{{a}_{11}}\] element of \[{{1}^{st}}\] matrix with the \[{{a}_{11}}\] element of \[{{2}^{nd}}\] matrix. Do the same for \[{{a}_{12}}\] elements of A and B matrices. Now, use the property of multiplication of a scalar to multiply 2 with the resultant matrix A + B. Multiply 2 with each element inside the matrix to get the answer.

Complete step by step answer:
We have been given two matrices: -
\[\begin{align}
  & A=\left[ \begin{matrix}
   4 & 5 \\
\end{matrix} \right] \\
 & B=\left[ \begin{matrix}
   7 & -3 \\
\end{matrix} \right] \\
\end{align}\]
These can be said as row matrices because they have only one row. We know that, generally a matrix is represented as,
\[\Rightarrow M={{\left[ \begin{align}
  & {{a}_{11}}{{a}_{12}}{{a}_{13}}.......{{a}_{1m}} \\
 & {{a}_{21}}{{a}_{22}}{{a}_{23}}.......{{a}_{2m}} \\
 & ............................. \\
 & {{a}_{n1}}{{a}_{n2}}{{a}_{n3}}.......{{a}_{nm}} \\
\end{align} \right]}_{n\times m}}\]
Here, \[n\times m\] is read as ‘n cross m’ which is also called the order. Here, ‘n’ is the number of rows and ‘m’ is the number of columns.
Now, when we have two matrices M and N, then the addition property says that, element \[{{a}_{11}}\] of matrix M should be added to \[{{a}_{11}}\] element of matrix N. Similarly, \[{{a}_{12}}\] of M with \[{{a}_{12}}\] of N and so on.
So, addition of given matrices A and B will give,
\[A+B=\left[ \begin{matrix}
   4 & 5 \\
\end{matrix} \right]+\left[ \begin{matrix}
   7 & -3 \\
\end{matrix} \right]\]
Here, \[{{a}_{11}}\] elements of A and B are 4 and 7 respectively while \[{{a}_{12}}\] elements of A and B are 5 and -3 respectively.
\[\begin{align}
  & \Rightarrow A+B=\left[ \begin{matrix}
   4+7 & 5+\left( -3 \right) \\
\end{matrix} \right] \\
 & \Rightarrow A+B=\left[ \begin{matrix}
   11 & 2 \\
\end{matrix} \right] \\
\end{align}\]
Now, we have to multiply 2 with the resultant matrix A + B. So, using the property of multiplication of a scalar with a matrix which states that if a scalar ‘a’ is multiplied to a matrix then each element of that matrix is multiplied by ‘a’. Therefore, we have,
\[\begin{align}
  & \Rightarrow 2\left[ A+B \right]=2\times \left[ \begin{matrix}
   11 & 2 \\
\end{matrix} \right] \\
 & \Rightarrow 2\left( A+B \right)=\left[ \begin{matrix}
   22 & 4 \\
\end{matrix} \right] \\
\end{align}\]
Hence, option (c) is the correct answer.

Note:
One may note that in the above solution we have added the matrices A and B first and then multiplied 2 with the resultant matrix, A + B. There is another method also, to get the answer. We can find the value of 2A and 2B first by multiplying 2 with A and B respectively. Then, we perform the addition operation to get the result. We will note that we are getting the same answer. This can be proved mathematically. Since, 2A + 2B = 2 (A + B).