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Algebra lineare Esempi
14x-21y-7z=1014x−21y−7z=10 , -4x+2y-2z=4−4x+2y−2z=4 , 56x-21y+7z=556x−21y+7z=5
Passaggio 1
Trova AX=BAX=B dal sistema di equazioni.
[14-21-7-42-256-217]⋅[xyz]=[1045]⎡⎢⎣14−21−7−42−256−217⎤⎥⎦⋅⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣1045⎤⎥⎦
Passaggio 2
Passaggio 2.1
Find the determinant.
Passaggio 2.1.1
Choose the row or column with the most 00 elements. If there are no 00 elements choose any row or column. Multiply every element in row 11 by its cofactor and add.
Passaggio 2.1.1.1
Consider the corresponding sign chart.
|+-+-+-+-+|∣∣
∣∣+−+−+−+−+∣∣
∣∣
Passaggio 2.1.1.2
The cofactor is the minor with the sign changed if the indices match a -− position on the sign chart.
Passaggio 2.1.1.3
The minor for a11a11 is the determinant with row 11 and column 11 deleted.
|2-2-217|∣∣∣2−2−217∣∣∣
Passaggio 2.1.1.4
Multiply element a11a11 by its cofactor.
14|2-2-217|14∣∣∣2−2−217∣∣∣
Passaggio 2.1.1.5
The minor for a12a12 is the determinant with row 11 and column 22 deleted.
|-4-2567|∣∣∣−4−2567∣∣∣
Passaggio 2.1.1.6
Multiply element a12a12 by its cofactor.
21|-4-2567|21∣∣∣−4−2567∣∣∣
Passaggio 2.1.1.7
The minor for a13a13 is the determinant with row 11 and column 33 deleted.
|-4256-21|∣∣∣−4256−21∣∣∣
Passaggio 2.1.1.8
Multiply element a13a13 by its cofactor.
-7|-4256-21|−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.1.9
Add the terms together.
14|2-2-217|+21|-4-2567|-7|-4256-21|14∣∣∣2−2−217∣∣∣+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
14|2-2-217|+21|-4-2567|-7|-4256-21|14∣∣∣2−2−217∣∣∣+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.2
Calcola |2-2-217|∣∣∣2−2−217∣∣∣.
Passaggio 2.1.2.1
È possibile trovare il determinante di una matrice 2×22×2 usando la formula |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb.
14(2⋅7-(-21⋅-2))+21|-4-2567|-7|-4256-21|14(2⋅7−(−21⋅−2))+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.2.2
Semplifica il determinante.
Passaggio 2.1.2.2.1
Semplifica ciascun termine.
Passaggio 2.1.2.2.1.1
Moltiplica 22 per 77.
14(14-(-21⋅-2))+21|-4-2567|-7|-4256-21|14(14−(−21⋅−2))+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.2.2.1.2
Moltiplica -(-21⋅-2)−(−21⋅−2).
Passaggio 2.1.2.2.1.2.1
Moltiplica -21−21 per -2−2.
14(14-1⋅42)+21|-4-2567|-7|-4256-21|14(14−1⋅42)+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.2.2.1.2.2
Moltiplica -1−1 per 4242.
14(14-42)+21|-4-2567|-7|-4256-21|14(14−42)+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
14(14-42)+21|-4-2567|-7|-4256-21|14(14−42)+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
14(14-42)+21|-4-2567|-7|-4256-21|14(14−42)+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.2.2.2
Sottrai 4242 da 1414.
14⋅-28+21|-4-2567|-7|-4256-21|14⋅−28+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
14⋅-28+21|-4-2567|-7|-4256-21|14⋅−28+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
14⋅-28+21|-4-2567|-7|-4256-21|14⋅−28+21∣∣∣−4−2567∣∣∣−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.3
Calcola |-4-2567|∣∣∣−4−2567∣∣∣.
Passaggio 2.1.3.1
È possibile trovare il determinante di una matrice 2×22×2 usando la formula |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb.
14⋅-28+21(-4⋅7-56⋅-2)-7|-4256-21|14⋅−28+21(−4⋅7−56⋅−2)−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.3.2
Semplifica il determinante.
Passaggio 2.1.3.2.1
Semplifica ciascun termine.
Passaggio 2.1.3.2.1.1
Moltiplica -4−4 per 77.
14⋅-28+21(-28-56⋅-2)-7|-4256-21|14⋅−28+21(−28−56⋅−2)−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.3.2.1.2
Moltiplica -56−56 per -2−2.
14⋅-28+21(-28+112)-7|-4256-21|14⋅−28+21(−28+112)−7∣∣∣−4256−21∣∣∣
14⋅-28+21(-28+112)-7|-4256-21|14⋅−28+21(−28+112)−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.3.2.2
Somma -28−28 e 112112.
14⋅-28+21⋅84-7|-4256-21|14⋅−28+21⋅84−7∣∣∣−4256−21∣∣∣
14⋅-28+21⋅84-7|-4256-21|14⋅−28+21⋅84−7∣∣∣−4256−21∣∣∣
14⋅-28+21⋅84-7|-4256-21|14⋅−28+21⋅84−7∣∣∣−4256−21∣∣∣
Passaggio 2.1.4
Calcola |-4256-21|∣∣∣−4256−21∣∣∣.
Passaggio 2.1.4.1
È possibile trovare il determinante di una matrice 2×22×2 usando la formula |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb.
14⋅-28+21⋅84-7(-4⋅-21-56⋅2)14⋅−28+21⋅84−7(−4⋅−21−56⋅2)
Passaggio 2.1.4.2
Semplifica il determinante.
Passaggio 2.1.4.2.1
Semplifica ciascun termine.
Passaggio 2.1.4.2.1.1
Moltiplica -4−4 per -21−21.
14⋅-28+21⋅84-7(84-56⋅2)14⋅−28+21⋅84−7(84−56⋅2)
Passaggio 2.1.4.2.1.2
Moltiplica -56−56 per 22.
14⋅-28+21⋅84-7(84-112)14⋅−28+21⋅84−7(84−112)
14⋅-28+21⋅84-7(84-112)14⋅−28+21⋅84−7(84−112)
Passaggio 2.1.4.2.2
Sottrai 112112 da 8484.
14⋅-28+21⋅84-7⋅-2814⋅−28+21⋅84−7⋅−28
14⋅-28+21⋅84-7⋅-2814⋅−28+21⋅84−7⋅−28
14⋅-28+21⋅84-7⋅-2814⋅−28+21⋅84−7⋅−28
Passaggio 2.1.5
Semplifica il determinante.
Passaggio 2.1.5.1
Semplifica ciascun termine.
Passaggio 2.1.5.1.1
Moltiplica 1414 per -28−28.
-392+21⋅84-7⋅-28−392+21⋅84−7⋅−28
Passaggio 2.1.5.1.2
Moltiplica 2121 per 8484.
-392+1764-7⋅-28−392+1764−7⋅−28
Passaggio 2.1.5.1.3
Moltiplica -7−7 per -28−28.
-392+1764+196−392+1764+196
-392+1764+196−392+1764+196
Passaggio 2.1.5.2
Somma -392−392 e 17641764.
1372+1961372+196
Passaggio 2.1.5.3
Somma 13721372 e 196196.
15681568
15681568
15681568
Passaggio 2.2
Since the determinant is non-zero, the inverse exists.
Passaggio 2.3
Set up a 3×63×6 matrix where the left half is the original matrix and the right half is its identity matrix.
[14-21-7100-42-201056-217001]⎡⎢⎣14−21−7100−42−201056−217001⎤⎥⎦
Passaggio 2.4
Trova la forma ridotta a scala per righe di Echelon.
Passaggio 2.4.1
Multiply each element of R1R1 by 114114 to make the entry at 1,11,1 a 11.
Passaggio 2.4.1.1
Multiply each element of R1R1 by 114114 to make the entry at 1,11,1 a 11.
[1414-2114-714114014014-42-201056-217001]⎡⎢
⎢⎣1414−2114−714114014014−42−201056−217001⎤⎥
⎥⎦
Passaggio 2.4.1.2
Semplifica R1R1.
[1-32-1211400-42-201056-217001]⎡⎢
⎢⎣1−32−1211400−42−201056−217001⎤⎥
⎥⎦
[1-32-1211400-42-201056-217001]⎡⎢
⎢⎣1−32−1211400−42−201056−217001⎤⎥
⎥⎦
Passaggio 2.4.2
Perform the row operation R2=R2+4R1R2=R2+4R1 to make the entry at 2,12,1 a 00.
Passaggio 2.4.2.1
Perform the row operation R2=R2+4R1R2=R2+4R1 to make the entry at 2,12,1 a 00.
[1-32-1211400-4+4⋅12+4(-32)-2+4(-12)0+4(114)1+4⋅00+4⋅056-217001]⎡⎢
⎢
⎢⎣1−32−1211400−4+4⋅12+4(−32)−2+4(−12)0+4(114)1+4⋅00+4⋅056−217001⎤⎥
⎥
⎥⎦
Passaggio 2.4.2.2
Semplifica R2R2.
[1-32-12114000-4-4271056-217001]⎡⎢
⎢⎣1−32−12114000−4−4271056−217001⎤⎥
⎥⎦
[1-32-12114000-4-4271056-217001]⎡⎢
⎢⎣1−32−12114000−4−4271056−217001⎤⎥
⎥⎦
Passaggio 2.4.3
Perform the row operation R3=R3-56R1R3=R3−56R1 to make the entry at 3,13,1 a 00.
Passaggio 2.4.3.1
Perform the row operation R3=R3-56R1R3=R3−56R1 to make the entry at 3,13,1 a 00.
[1-32-12114000-4-4271056-56⋅1-21-56(-32)7-56(-12)0-56(114)0-56⋅01-56⋅0]⎡⎢
⎢
⎢
⎢⎣1−32−12114000−4−4271056−56⋅1−21−56(−32)7−56(−12)0−56(114)0−56⋅01−56⋅0⎤⎥
⎥
⎥
⎥⎦
Passaggio 2.4.3.2
Semplifica R3R3.
[1-32-12114000-4-4271006335-401]⎡⎢
⎢⎣1−32−12114000−4−4271006335−401⎤⎥
⎥⎦
[1-32-12114000-4-4271006335-401]⎡⎢
⎢⎣1−32−12114000−4−4271006335−401⎤⎥
⎥⎦
Passaggio 2.4.4
Multiply each element of R2R2 by -14−14 to make the entry at 2,22,2 a 11.
Passaggio 2.4.4.1
Multiply each element of R2R2 by -14−14 to make the entry at 2,22,2 a 11.
[1-32-1211400-14⋅0-14⋅-4-14⋅-4-14⋅27-14⋅1-14⋅006335-401]⎡⎢
⎢⎣1−32−1211400−14⋅0−14⋅−4−14⋅−4−14⋅27−14⋅1−14⋅006335−401⎤⎥
⎥⎦
Passaggio 2.4.4.2
Semplifica R2R2.
[1-32-1211400011-114-14006335-401]⎡⎢
⎢⎣1−32−1211400011−114−14006335−401⎤⎥
⎥⎦
[1-32-1211400011-114-14006335-401]⎡⎢
⎢⎣1−32−1211400011−114−14006335−401⎤⎥
⎥⎦
Passaggio 2.4.5
Perform the row operation R3=R3-63R2R3=R3−63R2 to make the entry at 3,23,2 a 00.
Passaggio 2.4.5.1
Perform the row operation R3=R3-63R2R3=R3−63R2 to make the entry at 3,2 a 0.
[1-32-1211400011-114-1400-63⋅063-63⋅135-63⋅1-4-63(-114)0-63(-14)1-63⋅0]
Passaggio 2.4.5.2
Semplifica R3.
[1-32-1211400011-114-14000-28126341]
[1-32-1211400011-114-14000-28126341]
Passaggio 2.4.6
Multiply each element of R3 by -128 to make the entry at 3,3 a 1.
Passaggio 2.4.6.1
Multiply each element of R3 by -128 to make the entry at 3,3 a 1.
[1-32-1211400011-114-140-128⋅0-128⋅0-128⋅-28-128⋅12-128⋅634-128⋅1]
Passaggio 2.4.6.2
Semplifica R3.
[1-32-1211400011-114-140001-156-916-128]
[1-32-1211400011-114-140001-156-916-128]
Passaggio 2.4.7
Perform the row operation R2=R2-R3 to make the entry at 2,3 a 0.
Passaggio 2.4.7.1
Perform the row operation R2=R2-R3 to make the entry at 2,3 a 0.
[1-32-12114000-01-01-1-114+156-14+9160+128001-156-916-128]
Passaggio 2.4.7.2
Semplifica R2.
[1-32-1211400010-356516128001-156-916-128]
[1-32-1211400010-356516128001-156-916-128]
Passaggio 2.4.8
Perform the row operation R1=R1+12R3 to make the entry at 1,3 a 0.
Passaggio 2.4.8.1
Perform the row operation R1=R1+12R3 to make the entry at 1,3 a 0.
[1+12⋅0-32+12⋅0-12+12⋅1114+12(-156)0+12(-916)0+12(-128)010-356516128001-156-916-128]
Passaggio 2.4.8.2
Semplifica R1.
[1-320116-932-156010-356516128001-156-916-128]
[1-320116-932-156010-356516128001-156-916-128]
Passaggio 2.4.9
Perform the row operation R1=R1+32R2 to make the entry at 1,2 a 0.
Passaggio 2.4.9.1
Perform the row operation R1=R1+32R2 to make the entry at 1,2 a 0.
[1+32⋅0-32+32⋅10+32⋅0116+32(-356)-932+32⋅516-156+32⋅128010-356516128001-156-916-128]
Passaggio 2.4.9.2
Semplifica R1.
[100-156316128010-356516128001-156-916-128]
[100-156316128010-356516128001-156-916-128]
[100-156316128010-356516128001-156-916-128]
Passaggio 2.5
The right half of the reduced row echelon form is the inverse.
[-156316128-356516128-156-916-128]
[-156316128-356516128-156-916-128]
Passaggio 3
Moltiplica a sinistra entrambi i lati dell'equazione della matrice per la matrice inversa.
([-156316128-356516128-156-916-128]⋅[14-21-7-42-256-217])⋅[xyz]=[-156316128-356516128-156-916-128]⋅[1045]
Passaggio 4
Qualsiasi matrice moltiplicata per il suo inverso è sempre uguale a 1. A⋅A-1=1.
[xyz]=[-156316128-356516128-156-916-128]⋅[1045]
Passaggio 5
Passaggio 5.1
Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is 3×3 and the second matrix is 3×1.
Passaggio 5.2
Moltiplica ogni riga nella prima matrice per ogni colonna nella seconda matrice.
[-156⋅10+316⋅4+128⋅5-356⋅10+516⋅4+128⋅5-156⋅10-916⋅4-128⋅5]
Passaggio 5.3
Semplifica ogni elemento della matrice moltiplicando tutte le espressioni.
[342528-7328]
[342528-7328]
Passaggio 6
Semplifica il lato destro e sinistro.
[xyz]=[342528-7328]
Passaggio 7
Trova la soluzione.
x=34
y=2528
z=-7328