Determinant of a square Matrix

Solve Using :

By Expanding

Solution

Determinant : -82392

See below for the detailed solution.

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******** Solving By Laplace Transformation *******
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Solving Determinant, Expanding by Row: 1
Determinant = 20·det[D0]-6·det[D1]+5·det[D2]-8·det[D3]     Equation # 1
Where : 
D0 = 
 [  -3    2    12   |
 |   18   22   0    |
 |   6   -4    40   ]

D1 = 
 [   0    2    12   |
 |   21   22   0    |
 |   9   -4    40   ]

D2 = 
 [   0   -3    12   |
 |   21   18   0    |
 |   9    6    40   ]

D3 = 
 [   0   -3    2    |
 |   21   18   22   |
 |   9    6   -4    ]

Solving Det[D0], Expanding by Row: 1
   Det[D0] = -3·det[D4]-2·det[D5]+12·det[D6]     Equation # 2
   Where : 
   D4 = 
    [   22   0    |
    |  -4    40   ]

D5 = 
    [   18   0    |
    |   6    40   ]

D6 = 
    [   18   22   |
    |   6   -4    ]

Det[D4] = (22 · 40) - (0 · -4)
       = 880
      Det[D5] = (18 · 40) - (0 · 6)
       = 720
      Det[D6] = (18 · -4) - (22 · 6)
       = -204
   Substituting D4, D5, D6, in Equation:2, we get
   Det[D0] = -6528

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   Solving Det[D1], Expanding by Row: 1
   Det[D1] = 0·det[D7]-2·det[D8]+12·det[D9]
   Removing the zero entries, we get
   Det[D1] = -2·det[D8]+12·det[D9]     Equation # 3
   Where : 
   D8 = 
    [   21   0    |
    |   9    40   ]

D9 = 
    [   21   22   |
    |   9   -4    ]

Det[D8] = (21 · 40) - (0 · 9)
       = 840
      Det[D9] = (21 · -4) - (22 · 9)
       = -282
   Substituting D8, D9, in Equation:3, we get
   Det[D1] = -5064

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   Solving Det[D2], Expanding by Row: 1
   Det[D2] = 0·det[D10]+3·det[D11]+12·det[D12]
   Removing the zero entries, we get
   Det[D2] = 3·det[D11]+12·det[D12]     Equation # 4
   Where : 
   D11 = 
    [   21   0    |
    |   9    40   ]

D12 = 
    [   21   18   |
    |   9    6    ]

Det[D11] = (21 · 40) - (0 · 9)
       = 840
      Det[D12] = (21 · 6) - (18 · 9)
       = -36
   Substituting D11, D12, in Equation:4, we get
   Det[D2] = 2088

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   Solving Det[D3], Expanding by Row: 1
   Det[D3] = 0·det[D13]+3·det[D14]+2·det[D15]
   Removing the zero entries, we get
   Det[D3] = 3·det[D14]+2·det[D15]     Equation # 5
   Where : 
   D14 = 
    [   21   22   |
    |   9   -4    ]

D15 = 
    [   21   18   |
    |   9    6    ]

Det[D14] = (21 · -4) - (22 · 9)
       = -282
      Det[D15] = (21 · 6) - (18 · 9)
       = -36
   Substituting D14, D15, in Equation:5, we get
   Det[D3] = -918

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Substituting D0, D1, D2, D3, in Equation:1, we get
Determinant = -82392

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