Question

Please solve this matrix problem​

Answer 1

GIVEN :–

$\\ \longrightarrow\bf \:A = \left[\begin{array}{cc}1&2 \\ 4&2\end{array}\right]$

TO PROVE :–

$\\ \longrightarrow\bf |2A| = 4|A|$

SOLUTION :–

• Let take L.H.S. –

$\\ \: \: = \: \: \bf |2A|$

$\\ \: \: = \: \: \bf \bigg|2\left[\begin{array}{cc}1&2 \\ 4&2\end{array}\right] \bigg|$

$\\ \: \: = \: \: \bf \bigg|\left[\begin{array}{cc}1 \times 2&2 \times 2 \\ 4 \times 2&2 \times 2\end{array}\right] \bigg|$

$\\ \: \: = \: \: \bf\left| \begin{array}{cc}1 \times 2&2 \times 2 \\ 4 \times 2&2 \times 2\end{array}\right|$

$\\ \: \: = \: \: \bf\left| \begin{array}{cc}2&4\\8&4\end{array}\right|$

$\\ \: \: = \: \: \bf4 \times 2 - 8 \times 4$

$\\ \: \: = \: \: \bf8-32$

$\\ \: \: = \: \: \bf-24 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:- - - eq.(1)$

• Now let's take R.H.S. –

$\\ \: \: = \: \: \bf 4|A|$

$\\ \: \: = \: \: \bf 4 \left|\left[\begin{array}{cc}1&2 \\ 4&2\end{array}\right] \right|$

$\\ \: \: = \: \: \bf 4 \left|\begin{array}{cc}1&2 \\ 4&2\end{array} \right|$

$\\ \: \: = \: \: \bf 4(2 \times 1 - 4 \times 2)$

$\\ \: \: = \: \: \bf 4(2-8)$

$\\ \: \: = \: \: \bf 4(-6)$

$\\ \: \: = \: \: \bf - 24\: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \:- - -eq.(2)$

• By eq.(1) & eq.(2) –

$\\ \implies\bf\:L.H.S.= \:R.H.S.$

$\\ \implies\bf{ \green{\underbrace{\red{Hence \: \: Proved}}}}$