Question

Find the value of $b$ if
$\left[ \begin{matrix}
   a-b & 2a+c \\
   2a-b & 3c+d \\
\end{matrix} \right]=\left[ \begin{matrix}
   -1 & 5 \\
   0 & 13 \\
\end{matrix} \right]$

Answer 1

Hint: We start solving this problem by equating the corresponding terms on both the sides. After equating the corresponding terms on both sides, we get a system of four linear equations. Then we start solving them to eliminate other variables and to get the value of $b$.

Complete step by step answer:
Let us consider the given matrices,
$\left[ \begin{matrix}
   a-b & 2a+c \\
   2a-b & 3c+d \\
\end{matrix} \right]=\left[ \begin{matrix}
   -1 & 5 \\
   0 & 13 \\
\end{matrix} \right]$
Now, let us equate the corresponding terms on the both sides of the matrices.
Then by equating the element in the first row and first column we get,
$a-b=-1.................\left( 1 \right)$
Then by equating the element in the first row and second column we get,
$2a+c=5...............\left( 2 \right)$
Similarly, by equating the element in the second row and first column we get,
$2a-b=0...............\left( 3 \right)$
Then by equating the element in the second row and second column we get,
$3c+d=13.............\left( 4 \right)$
Now, let us consider the equations (1) and (3).
We first multiply the equation (1) with 2 on both sides to get $2a$ so that we cancel that term while subtracting it from equation (3). By doing so, we get
$\begin{align}
  & 2\left( a-b \right)=2\left( -1 \right) \\
 & \Rightarrow 2a-2b=-2............\left( 5 \right) \\
\end{align}$
Now, we subtract equation (5) from equation (3). Then we get,
$\begin{align}
  & \left( 2a-b \right)-\left( 2a-2b \right)=0-\left( -2 \right) \\
 & \Rightarrow 2a-b-2a+2b=0+2 \\
 & \Rightarrow -b+2b=2 \\
 & \Rightarrow b=2 \\
\end{align}$
Hence, the value of $b$ is 2.

So, the correct answer is 2.

Note: Here one might solve all the equations and find the respective values of c, d and a and then b. but it is a long way of solving the problem. As we are required to find the value of b, we need to use the equations containing b first, and use them to solve and find the value b. Here in this problem we can see that we have two equations with variables a and b. So, they are enough to find the value of b.