Question
If $\left(\begin{array}{lll}2 & 1 & 3\end{array}\right)\left(\begin{array}{ccc}-1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right)\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)=A$, then write the order of matrix $A$.
✓
Answer
We have,
$A=\left(\begin{array}{lll}2 & 1 & 3\end{array}\right)\left(\begin{array}{ccc}-1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right)\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$
$=\left(\begin{array}{lll}-2-1 & 1+3 & -2+3\end{array}\right)\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$
$=\left(\begin{array}{lll}-3 & 4 & 1\end{array}\right)\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$
$=(-3-1)=(-4)_{1 \times 1}$
$\therefore$ Order of matrix $A$ is $1 \times 1.$
OR
Let $P=\left(\begin{array}{lll}2 & 1 & 3\end{array}\right), Q=\left(\begin{array}{ccc}-1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right)$ and $R=\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$
order of $PQ =1 \times 3$
order of $PQR(A)=1\times1$
$A=\left(\begin{array}{lll}2 & 1 & 3\end{array}\right)\left(\begin{array}{ccc}-1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right)\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$
$=\left(\begin{array}{lll}-2-1 & 1+3 & -2+3\end{array}\right)\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$
$=\left(\begin{array}{lll}-3 & 4 & 1\end{array}\right)\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$
$=(-3-1)=(-4)_{1 \times 1}$
$\therefore$ Order of matrix $A$ is $1 \times 1.$
OR
Let $P=\left(\begin{array}{lll}2 & 1 & 3\end{array}\right), Q=\left(\begin{array}{ccc}-1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right)$ and $R=\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$
order of $PQ =1 \times 3$
order of $PQR(A)=1\times1$
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