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Find the value of $a$ if $\left[ {\begin{array}{*{20}{c}}
  {a - b}&{2a + c} \\
  {2a - b}&{3c + d}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  { - 1}&5 \\
  0&{13}
\end{array}} \right]$

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Hint: Equate the corresponding elements of both matrices.

We have to find the value of $a$, hence we are able to form 2 linear equations having the same variables with one variable as $a$ then simply solve them by using elimination method to find the required solution.

Compare the corresponding matrix elements as the two matrices given are equal.

Comparing ${A_{11}}$ of left matrix with ${A_{11}}$of right matrix we get
$a - b = - 1$……………………………… (1)
Comparing ${A_{21}}$of left matrix with ${A_{21}}$of right matrix we get
$2a - b = 0$………………………………. (2)
Let’s subtract equations (1) and (2) we get
$a - b - 2a + b = - 1 - 0$
$ \Rightarrow - a = - 1$
Hence the value of $a = 1$

Note- Whenever we come across such problems the only key concept that we need to follow is that we try to make linear equations in two variables with one variable as the required quantity that is to be found.
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