Determinants and Matrices — Maths STD 11 Science — Question

i.e, to prove $A^2-A B+B A-B^2=A^2-B^2$

i.e, to prove $-\mathrm{AB}+\mathrm{BA}=0$,

i.e., to prove $A B-B A$.

$\mathrm{AB}=\left[\begin{array}{cc}3 & 4 \\ -4 & 3\end{array}\right]\left[\begin{array}{cc}2 & 1 \\ -1 & 2\end{array}\right]$

$=\left[\begin{array}{cc}6-4 & 3+8 \\ -8-3 & -4+6\end{array}\right]$

$=\left[\begin{array}{cc}2 & 11 \\ -11 & 2\end{array}\right]$

$\ldots$..(i)

$\mathbf{B A}=\left[\begin{array}{cc}2 & 1 \\ -1 & 2\end{array}\right]\left[\begin{array}{cc}3 & 4 \\ -4 & 3\end{array}\right]$

$=\left[\begin{array}{cc}6-4 & 8+3 \\ -3-8 & -4+6\end{array}\right]$

$=\left[\begin{array}{cc}2 & 11 \\ -11 & 2\end{array}\right]$

$\ldots$..(ii)

From (i) and (ii), we get AB = BA