Question

If $\left[ {\begin{array}{*{20}{c}}
  2&{ - 3} \\
  4&0
\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}
  a&c \\
  b&d
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1&4 \\
  2&{ - 5}
\end{array}} \right]$ , then what is the value of $\left( {a,b,c,d} \right)$ ?
A. $\left( {1,6,2,5} \right)$
B. $\left( {1,2,7,5} \right)$
C. $\left( {1,2, - 7,5} \right)$
D. $\left( { - 1, - 2,7, - 5} \right)$

Answer 1

Hint: Perform Matrix Subtraction and evaluate the value of the Left-Hand side of the equation. Then, equate all the elements of the matrix obtained with the matrix given on the Right-Hand side. Evaluate the values of the unknown variables from the obtained equations.

Complete step by step Solution:
Given equation:
$\left[ {\begin{array}{*{20}{c}}
  2&{ - 3} \\
  4&0
\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}
  a&c \\
  b&d
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1&4 \\
  2&{ - 5}
\end{array}} \right]$
Performing matrix Subtraction of the Left-Hand side,
$\left[ {\begin{array}{*{20}{c}}
  {2 - a}&{ - 3 - c} \\
  {4 - b}&{0 - d}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1&4 \\
  2&{ - 5}
\end{array}} \right]$
Equating the elements of both the matrices,
$2 - a = 1$
$ - 3 - c = 4$
$4 - b = 2$
$0 - d = - 5$
From the above four equations, we get:
$a = 1$
$c = - 7$
$b = 2$
$d = 5$
Therefore, $\left( {a,b,c,d} \right) = \left( {1,2, - 7,5} \right)$

Hence, the correct option is (C).

Note:Two matrices $A$ and $B$ are said to be equal to each other when they both are of the same order and when every element of matrix $A$ is equal to the corresponding elements of matrix $B$.