Question

Find the row operation of the matrix: -
\[B=\left[ \begin{matrix}
   1 \\
   2 \\
\end{matrix}\begin{matrix}
   -1 \\
   5 \\
\end{matrix}\begin{matrix}
   3 \\
   4 \\
\end{matrix} \right],{{R}_{1}}\to {{R}_{1}}-{{R}_{2}}\]

Answer 1

Hint: First understand what is a row or column of matrix. In row 1, subtract the numbers present in row 1 with the numbers present in row 2 to find the value of the matrix operation, \[{{R}_{1}}\to {{R}_{1}}-{{R}_{2}}\].

Complete step by step answer:
We have been asked to find the row operation: - \[{{R}_{1}}\to {{R}_{1}}-{{R}_{2}}\] of the provided matrix: - \[B=\left[ \begin{matrix}
   1 \\
   2 \\
\end{matrix}\begin{matrix}
   -1 \\
   5 \\
\end{matrix}\begin{matrix}
   3 \\
   4 \\
\end{matrix} \right]\]. First, let us define a matrix and its rows and columns.
In mathematics, a matrix is a rectangular array or table of numbers, symbols or expressions arranged in rows and columns. Rows are horizontal lines in which numbers are arranged whereas columns are vertical lines in which numbers are arranged. Let us take the above example: -
\[B=\left[ \begin{matrix}
   1 \\
   2 \\
\end{matrix}\begin{matrix}
   -1 \\
   5 \\
\end{matrix}\begin{matrix}
   3 \\
   4 \\
\end{matrix} \right]\]
Here, there are two horizontal lines, so there are two rows. First one contains 1, -1 and 3 as its elements while second row contains 2, 5 and 4 as its elements.
Now, we can see that there are three vertical lines, so there are three columns. Column 1 contains 1 and 2, column 2 contains -1 and 5 and column 3 contains 3 and 4 as its elements.
We denote any row and column by \[{{R}_{n}}\] and \[{{C}_{n}}\] respectively, where ‘n’ is the \[{{n}^{th}}\] row or column.
Now, let us come to the question. We have to find: - \[{{R}_{1}}\to {{R}_{1}}-{{R}_{2}}\], which is a row operation. If it would have been \[{{C}_{1}}\to {{C}_{1}}-{{C}_{2}}\] then it would have been said as column operation.
Now, \[{{R}_{1}}\to {{R}_{1}}-{{R}_{2}}\]states that we have to change the elements of \[{{R}_{1}}\] with the elements obtained by the arithmetic operation, \[{{R}_{1}}-{{R}_{2}}\]. That means, elements of \[{{R}_{2}}\] is subtracted from the corresponding elements of \[{{R}_{1}}\]. Element 2 will be subtracted from element 1, element 5 from element 1 and element 4 from element 3.
\[\begin{align}
  & \Rightarrow \left[ \begin{matrix}
   1 \\
   2 \\
\end{matrix}\begin{matrix}
   -1 \\
   5 \\
\end{matrix}\begin{matrix}
   3 \\
   4 \\
\end{matrix} \right],{{R}_{1}}\to {{R}_{1}}-{{R}_{2}} \\
 & \Rightarrow \left[ \begin{matrix}
   1-2 \\
   2 \\
\end{matrix}\begin{matrix}
   -1-5 \\
   5 \\
\end{matrix}\begin{matrix}
   3-4 \\
   4 \\
\end{matrix} \right] \\
 & \Rightarrow \left[ \begin{matrix}
   -1 \\
   2 \\
\end{matrix}\begin{matrix}
   -6 \\
   5 \\
\end{matrix}\begin{matrix}
   -1 \\
   4 \\
\end{matrix} \right] \\
\end{align}\]
Which is the required matrix and our answer.

Note: One must not get confused in the rows and columns of a matrix otherwise you will get a wrong answer. Also, note that sometimes the order of a matrix is given as \[m\times n\] read as ‘m cross n’. It denotes nothing but rows and columns of a matrix. It says that a matrix has m rows and n columns.