Find A B, if A= [ ccc 1 & 2 & 3 1 & -2 & -3 ] and B= [ cc 1 & -1 … - Vidyadip

Question

Find $A B$, if $A=\left[\begin{array}{ccc}1 & 2 & 3 \\ 1 & -2 & -3\end{array}\right]$ and $B=\left[\begin{array}{cc}1 & -1 \\ 1 & 2 \\ 1 & -2\end{array}\right]$ Examine whether $A B$ has inverse or not.

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Answer

$A B=\left[\begin{array}{rrr}1 & 2 & 3 \\ 1 & -2 & -3\end{array}\right] \times\left[\begin{array}{rr}1 & -1 \\ 1 & 2 \\ 1 & -2\end{array}\right]$
$=\left[\begin{array}{rr}1(1)+2(1)+3(1) & 1(-1)+2(2)+3(-2) \\ 1(1)+(-2)(1)+(-3)(1) & 1(-1)+(-2)(2)+(-3)(-2)\end{array}\right]$
$=\left[\begin{array}{rr}1+2+3 & -1+4-6 \\ 1-2-3 & -1-4+6\end{array}\right]$
$=\left[\begin{array}{rr}6 & -3 \\ -4 & 1\end{array}\right] \\ \therefore|A B|=\left|\begin{array}{rr}6 & -3 \\ -4 & 1\end{array}\right|=6-12=-6 \neq 0$
$\therefore A$ is a non$-$singular matrix.
Hence, $(AB)^{-1}$ exist.

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