# How do you find the determinant of ((44, 32, 18, 6), (-2, 10, 15, 5), (21, 12, -12, 4), (-8, -16, 4, 9))?

May 1, 2016

$- 148 , 116$

#### Explanation:

$\left[\begin{matrix}44 & 32 & 18 & \textcolor{v i o \le t}{6} \\ - 2 & 10 & 15 & \textcolor{v i o \le t}{5} \\ 21 & 12 & - 12 & \textcolor{v i o \le t}{4} \\ - 8 & - 16 & 4 & \textcolor{v i o \le t}{9}\end{matrix}\right] =$

=6*(-1)^(1+4)[[-2,10,15],[color(teal)(21),color(teal)(12),color(teal)(-12)],[color(magenta)(-8),color(magenta)(-16),color(magenta)(4)]] +5*(-1)^(2+4)[[color(blue)(44),color(blue)(32),color(blue)(18)],[color(teal)(21),color(teal)(12),color(teal)(-12)],[color(magenta)(-8),color(magenta)(-16),color(magenta)(4)]]+4*(-1)^(3+4)[[color(blue)(44),color(blue)(32),color(blue)(18)],[-2,10,15],[color(magenta)(-8),color(magenta)(-16),color(magenta)(4)]]+9*(-1)^(4+4)[[color(blue)(44),color(blue)(32),color(blue)(18)],[-2,10,15],[color(magenta)(21),color(magenta)(12),color(magenta)(-12)]]=

=-6*3*4[[-2,10,15],[color(teal)(7),color(teal)(4),color(teal)(-4)],[color(magenta)(-2),color(magenta)(-4),color(magenta)(1)]] +5*2*3*4[[22,16,9],[color(teal)(7),color(teal)(4),color(teal)(-4)],[color(magenta)(-2),color(magenta)(-4),color(magenta)(1)]]-4*2*4[[22,16,9],[color(teal)(-2),color(teal)(10),color(teal)(15)],[color(magenta)(-2),color(magenta)(-4),color(magenta)(1)]]+9*2*3[[22,color(orange)(16),color(red)(9)],[-2,color(orange)(10),color(red)(15)],[7,color(orange)(4),color(red)(-4)]]=

=-72[[-2,color(violet)(10),15],[5,color(violet)(0),-3],[-2,color(violet)(-4),1]] +120[[22,color(violet)(16),9],[5,color(violet)(0),-3],[-2,color(violet)(-4),1]]-32[[color(violet)(22),16,9],[0,color(blue)(14),color(blue)(14)],[color(violet)(-2),-4,1]]+54[[22,color(violet)(25),9],[-2,color(violet)(25),15],[7,color(violet)(0),-4]]=

=-72*2[[-2,5,15],[5,0,-3],[-2,-2,1]] +120*4[[22,4,9],[5,0,-3],[-2,-1,1]]-32*2*14[[11,16,9],[0,1,1],[-1,-4,1]]+54*25[[22,1,9],[-2,1,15],[7,0,-4]]=

$= - 144 \left[30 - 150 - \left(- 12 + 25\right)\right] + 480 \left[24 - 45 - \left(66 + 20\right)\right] - 896 \left[11 - 16 - \left(- 9 - 44\right)\right] + 1350 \left[- 88 + 105 - \left(63 + 8\right)\right]$
$= - 144 \left[- 120 - 13\right] + 480 \left[- 21 - 86\right] - 896 \left[- 5 + 53\right] + 1350 \left[17 - 71\right]$
$= 144 \cdot 133 - 480 \cdot 107 - 896 \cdot 48 - 1350 \cdot 54$
$= 19152 - 51360 - 43008 - 72900 = - 148116$