# Find the determinants in Exercise , assuming that begin{vmatrix}a & b &c d&e&fg&h&i end{vmatrix}=4 begin{vmatrix}2a & 2b &2c d&e&fg&h&i end{vmatrix}

Find the determinants in Exercise , assuming that
$|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}|=4$
$|\begin{array}{ccc}2a& 2b& 2c\\ d& e& f\\ g& h& i\end{array}|$
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Macsen Nixon
Step 1
Determinant is used for solving linear equations. To find the inverse of a matrix determinant is used. It is computed only for square matrices. Square matrices are matrices whose number of rows and number of columns are equal.
Step 2
The given value of the determinant is $|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}|=4$ Multiplying any row or column of a determinant by a number x, multiplies the determinant by that number x. Using this property the value of the determinant $|\begin{array}{ccc}2a& 2b& 2c\\ d& e& f\\ g& h& i\end{array}|$ is calculated by taking the common term 2 from the first row and take the value 2 outside of the determinant, then it becomes 2 times of the determinant $|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}|$ , using the value given value find the asked determinant value as follows,
$|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}|=4$
$|\begin{array}{ccc}2a& 2b& 2c\\ d& e& f\\ g& h& i\end{array}|=|\begin{array}{ccc}2×a& 2×b& 2×c\\ d& e& f\\ g& h& i\end{array}|$
$=2|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}|$ Take the coomon term from first row outside of determinant.
$=8$
Hence, the value of the determinant asked in the question is equal to
$|\begin{array}{ccc}2a& 2b& 2c\\ d& e& f\\ g& h& i\end{array}|=8$
Jeffrey Jordon