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For the real numbers p, q, r, s, t, u, v, w, x, \(\begin{vmatrix} p & q & r \\ s & t & u \\ v & w & x \end{vmatrix} = -3.\) Find \(\begin{vmatrix} p & 2q & 5r + 4p \\ s & 2t & 5u + 4s \\ v & 2w & 5x + 4v \end{vmatrix}.\)
 

 Jun 14, 2020
 #1
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The determinant is (-3)(2)(5)(4) = -120.

 Jun 14, 2020
 #2
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By performing column operations and factoring out common factors whenever there is,

 

\(\begin{vmatrix} p & 2q & 5r + 4p \\ s & 2t & 5u + 4s \\ v & 2w & 5x + 4v \end{vmatrix} = 2 \begin{vmatrix} p & q & 5r \\ s & t & 5u \\ v & w & 5x \end{vmatrix} = 10 \begin{vmatrix} p & q & r \\ s & t & u \\ v & w & x \end{vmatrix} = 10(-3) = -30\)

 Jun 14, 2020

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