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Question Number 9287 by suci last updated on 28/Nov/16
find the determinant of the matrix below   determinant ((0,4,0,0,0),(0,0,0,2,0),(0,0,3,0,0),(0,0,0,0,1),(5,0,0,0,0))
$${find}\:{the}\:{determinant}\:{of}\:{the}\:{matrix}\:{below} \\ $$$$\begin{vmatrix}{\mathrm{0}}&{\mathrm{4}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{2}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{3}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{1}}\\{\mathrm{5}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\end{vmatrix} \\ $$
Answered by mrW last updated on 28/Nov/16
=−4× determinant ((0,0,2,0),(0,3,0,0),(0,0,0,1),(5,0,0,0))  =−4×2× determinant ((0,3,0),(0,0,1),(5,0,0))  =−4×2×(−3)× determinant ((0,1),(5,0))  =−4×2×(−3)×(−5×1)  =−120
$$=−\mathrm{4}×\begin{vmatrix}{\mathrm{0}}&{\mathrm{0}}&{\mathrm{2}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{3}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{1}}\\{\mathrm{5}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\end{vmatrix} \\ $$$$=−\mathrm{4}×\mathrm{2}×\begin{vmatrix}{\mathrm{0}}&{\mathrm{3}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{1}}\\{\mathrm{5}}&{\mathrm{0}}&{\mathrm{0}}\end{vmatrix} \\ $$$$=−\mathrm{4}×\mathrm{2}×\left(−\mathrm{3}\right)×\begin{vmatrix}{\mathrm{0}}&{\mathrm{1}}\\{\mathrm{5}}&{\mathrm{0}}\end{vmatrix} \\ $$$$=−\mathrm{4}×\mathrm{2}×\left(−\mathrm{3}\right)×\left(−\mathrm{5}×\mathrm{1}\right) \\ $$$$=−\mathrm{120} \\ $$
Answered by mrW last updated on 28/Nov/16
or  =− determinant ((0,0,0,0,1),(0,0,0,2,0),(0,0,3,0,0),(0,4,0,0,0),(5,0,0,0,0))  =−120
$$\mathrm{or} \\ $$$$=−\begin{vmatrix}{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{2}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{3}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{4}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{5}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\end{vmatrix} \\ $$$$=−\mathrm{120} \\ $$

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