Solve the following equations for

Question:

If $A=\left[\begin{array}{ll}1 & -1 \\ 2 & -1\end{array}\right], B=\left[\begin{array}{rr}a & 1 \\ b & -1\end{array}\right]$ and $(A+B)^{2}=A^{2}+B^{2}$, values of a and b are

(a) $a=4, b=1$

(b) $a=1, b=4$

(c) $a=0, b=4$

(d) $a=2, b=4$

Solution:

(b) $a=1, b=4$

Here,

$(A+B)^{2}=A^{2}+B^{2}$

$\Rightarrow A^{2}+A B+B A+B^{2}=A^{2}+B^{2}$

$\Rightarrow A B+B A=O$

$\Rightarrow A B=-B A$

$\Rightarrow\left[\begin{array}{ll}1 & -1 \\ 2 & -1\end{array}\right]\left[\begin{array}{cc}a & 1 \\ b & -1\end{array}\right]=-\left[\begin{array}{cc}a & 1 \\ b & -1\end{array}\right]\left[\begin{array}{cc}1 & -1 \\ 2 & -1\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}a-b & 2 \\ 2 a-b & 3\end{array}\right]=-\left[\begin{array}{cc}a+2 & -a-1 \\ b-2 & -b+1\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}a-b & 2 \\ 2 a-b & 3\end{array}\right]=\left[\begin{array}{cc}-a-2 & a+1 \\ b+2 & b-1\end{array}\right]$

The corresponding elements of two equal matrices are equal.

$\Rightarrow a+1=2$ and $b-1=3$

$\therefore a=1$ and $b=4$