Question

if A is a third order of matix and |A|=6 then what is the value of |4A|​

Answer 1

Answer: 64|A|

Given:

$\sf\text{A is a third order matrix and} \left| \begin{array}{c} A \end{array}\right|=6$

To find:

$\left| \begin{array}{c} 4A \end{array}\right|$

Explanation:

$\sf\text{Let A}=\left[\begin{array}{ccc}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{array}\right] _{3\times3}$

$\sf\text{Now,} \left| \begin{array}{c} 4A \end{array}\right|=$

$\sf 4A=4\left [\begin{array}{ccc}4a_1&4b_1&4c_1\\4a_2&4b_2&4c_2\\4a_3&4b_3&4c_3\end{array}\right]$

$\sf Also,$

$\sf |\begin{array}{c}4A\end{array}\right| |=\left |\begin{array}{ccc}4a_1&4b_1&4c_1\\4a_2&4b_2&4c_2\\4a_3&4b_3&4c_3\end{array}\right|$

$\sf\text{Taking out 4 as common from row 1, row2 , row 3, we get:}$

$\Rightarrow\sf 4\times4\times4\left |\begin{array}{ccc}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{array}\right|$

$\Rightarrow\sf 64\left |\begin{array}{ccc}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{array}\right|$

$\Rightarrow{\sf 64\left |\begin{array}{ccc}A_\end{array}\right|}$