Question

If $A$ and $B$ are two matrices such that $A = \left[ {\begin{array}{*{20}{c}}
  5&{ - 3} \\
  2&4
\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}
  6&{ - 4} \\
  3&6
\end{array}} \right]$ , then what is the value of $A - B$ ?
A. $\left[ {\begin{array}{*{20}{c}}
  {11}&{ - 7} \\
  5&{10}
\end{array}} \right]$
B. $\left[ {\begin{array}{*{20}{c}}
  { - 1}&1 \\
  { - 1}&{ - 2}
\end{array}} \right]$
C. $\left[ {\begin{array}{*{20}{c}}
  {11}&7 \\
  5&{ - 10}
\end{array}} \right]$
D. $\left[ {\begin{array}{*{20}{c}}
  {12}&{ - 7} \\
  5&{ - 10}
\end{array}} \right]$

Answer 1

Hint: Perform Matrix Subtraction and subtract each element of Matrix $B$ from the corresponding element of the matrix $A$ to get the desired value of the matrix which comes as a result of $A - B$.

Complete step by step Solution:
Given are two matrices, $A$ and $B$ such that:
$A = \left[ {\begin{array}{*{20}{c}}
  5&{ - 3} \\
  2&4
\end{array}} \right]$
And
$B = \left[ {\begin{array}{*{20}{c}}
  6&{ - 4} \\
  3&6
\end{array}} \right]$
Performing Matrix Subtraction,
$A - B = \left[ {\begin{array}{*{20}{c}}
  5&{ - 3} \\
  2&4
\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}
  6&{ - 4} \\
  3&6
\end{array}} \right]$
Subtracting each element of the matrix $B$ from matrix $A$,
$A - B = \left[ {\begin{array}{*{20}{c}}
  {5 - 6}&{ - 3 + 4} \\
  {2 - 3}&{4 - 6}
\end{array}} \right]$
On simplifying further, we get:
$A - B = \left[ {\begin{array}{*{20}{c}}
  { - 1}&1 \\
  { - 1}&{ - 2}
\end{array}} \right]$

Therefore, the correct option is (B).

Note: The commutative property does not hold for matrix subtraction, that is, $A - B \ne B - A$ . It also does not hold for matrix multiplication, that is, $AB \ne BA$. However, it does hold for matrix addition, that is, $A + B = B + A$.