Algebra lineare Esempi

$[\begin{array}{c} 6 & 3 & 2 & 4 & 0 \\ 9 & 0 & - 4 & 1 & 0 \\ 8 & - 5 & 6 & 7 & 1 \\ 3 & 0 & 0 & 0 & 0 \\ 4 & 2 & 3 & 2 & 0 \end{array}]$

Passaggio 1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $4$ by its cofactor and add.

Tocca per altri passaggi...

Passaggio 1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - & + \\ - & + & - & + & - \\ + & - & + & - & + \\ - & + & - & + & - \\ + & - & + & - & + \end{array} |$

Passaggio 1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Passaggio 1.3

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

$| \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} |$

Passaggio 1.4

Multiply element $a_{41}$ by its cofactor.

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} |$

Passaggio 1.5

The minor for $a_{42}$ is the determinant with row $4$ and column $2$ deleted.

$| \begin{array}{c} 6 & 2 & 4 & 0 \\ 9 & - 4 & 1 & 0 \\ 8 & 6 & 7 & 1 \\ 4 & 3 & 2 & 0 \end{array} |$

Passaggio 1.6

Multiply element $a_{42}$ by its cofactor.

$0 | \begin{array}{c} 6 & 2 & 4 & 0 \\ 9 & - 4 & 1 & 0 \\ 8 & 6 & 7 & 1 \\ 4 & 3 & 2 & 0 \end{array} |$

Passaggio 1.7

The minor for $a_{43}$ is the determinant with row $4$ and column $3$ deleted.

$| \begin{array}{c} 6 & 3 & 4 & 0 \\ 9 & 0 & 1 & 0 \\ 8 & - 5 & 7 & 1 \\ 4 & 2 & 2 & 0 \end{array} |$

Passaggio 1.8

Multiply element $a_{43}$ by its cofactor.

$0 | \begin{array}{c} 6 & 3 & 4 & 0 \\ 9 & 0 & 1 & 0 \\ 8 & - 5 & 7 & 1 \\ 4 & 2 & 2 & 0 \end{array} |$

Passaggio 1.9

The minor for $a_{44}$ is the determinant with row $4$ and column $4$ deleted.

$| \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} |$

Passaggio 1.10

Multiply element $a_{44}$ by its cofactor.

$0 | \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} |$

Passaggio 1.11

The minor for $a_{45}$ is the determinant with row $4$ and column $5$ deleted.

$| \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Passaggio 1.12

Multiply element $a_{45}$ by its cofactor.

$0 | \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Passaggio 1.13

Add the terms together.

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 2 & 4 & 0 \\ 9 & - 4 & 1 & 0 \\ 8 & 6 & 7 & 1 \\ 4 & 3 & 2 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 4 & 0 \\ 9 & 0 & 1 & 0 \\ 8 & - 5 & 7 & 1 \\ 4 & 2 & 2 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Passaggio 2

Moltiplica $0$ per $| \begin{array}{c} 6 & 2 & 4 & 0 \\ 9 & - 4 & 1 & 0 \\ 8 & 6 & 7 & 1 \\ 4 & 3 & 2 & 0 \end{array} |$ .

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} | + 0 + 0 | \begin{array}{c} 6 & 3 & 4 & 0 \\ 9 & 0 & 1 & 0 \\ 8 & - 5 & 7 & 1 \\ 4 & 2 & 2 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Passaggio 3

Moltiplica $0$ per $| \begin{array}{c} 6 & 3 & 4 & 0 \\ 9 & 0 & 1 & 0 \\ 8 & - 5 & 7 & 1 \\ 4 & 2 & 2 & 0 \end{array} |$ .

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} | + 0 + 0 + 0 | \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Passaggio 4

Moltiplica $0$ per $| \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} |$ .

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} | + 0 + 0 + 0 + 0 | \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Passaggio 5

Moltiplica $0$ per $| \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$ .

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} | + 0 + 0 + 0 + 0$

Passaggio 6

Calcola $| \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} |$ .

Tocca per altri passaggi...

Passaggio 6.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $4$ by its cofactor and add.

Tocca per altri passaggi...

Passaggio 6.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Passaggio 6.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Passaggio 6.1.3

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 0 & - 4 & 1 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$

Passaggio 6.1.4

Multiply element $a_{14}$ by its cofactor.

$0 | \begin{array}{c} 0 & - 4 & 1 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$

Passaggio 6.1.5

The minor for $a_{24}$ is the determinant with row $2$ and column $4$ deleted.

$| \begin{array}{c} 3 & 2 & 4 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$

Passaggio 6.1.6

Multiply element $a_{24}$ by its cofactor.

$0 | \begin{array}{c} 3 & 2 & 4 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$

Passaggio 6.1.7

The minor for $a_{34}$ is the determinant with row $3$ and column $4$ deleted.

$| \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ 2 & 3 & 2 \end{array} |$

Passaggio 6.1.8

Multiply element $a_{34}$ by its cofactor.

$- 1 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ 2 & 3 & 2 \end{array} |$

Passaggio 6.1.9

The minor for $a_{44}$ is the determinant with row $4$ and column $4$ deleted.

$| \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |$

Passaggio 6.1.10

Multiply element $a_{44}$ by its cofactor.

$0 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |$

Passaggio 6.1.11

Add the terms together.

$- 3 (0 | \begin{array}{c} 0 & - 4 & 1 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 2 & 4 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} | - 1 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ 2 & 3 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |) + 0 + 0 + 0 + 0$

Passaggio 6.2

Moltiplica $0$ per $| \begin{array}{c} 0 & - 4 & 1 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$ .

$- 3 (0 + 0 | \begin{array}{c} 3 & 2 & 4 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} | - 1 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ 2 & 3 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |) + 0 + 0 + 0 + 0$

Passaggio 6.3

Moltiplica $0$ per $| \begin{array}{c} 3 & 2 & 4 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$ .

$- 3 (0 + 0 - 1 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ 2 & 3 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |) + 0 + 0 + 0 + 0$

Passaggio 6.4

Moltiplica $0$ per $| \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |$ .

$- 3 (0 + 0 - 1 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ 2 & 3 & 2 \end{array} | + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5

Calcola $| \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ 2 & 3 & 2 \end{array} |$ .

Tocca per altri passaggi...

Passaggio 6.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $1$ by its cofactor and add.

Tocca per altri passaggi...

Passaggio 6.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Passaggio 6.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Passaggio 6.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - 4 & 1 \\ 3 & 2 \end{array} |$

Passaggio 6.5.1.4

Multiply element $a_{11}$ by its cofactor.

$3 | \begin{array}{c} - 4 & 1 \\ 3 & 2 \end{array} |$

Passaggio 6.5.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 2 & 4 \\ 3 & 2 \end{array} |$

Passaggio 6.5.1.6

Multiply element $a_{21}$ by its cofactor.

$0 | \begin{array}{c} 2 & 4 \\ 3 & 2 \end{array} |$

Passaggio 6.5.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 2 & 4 \\ - 4 & 1 \end{array} |$

Passaggio 6.5.1.8

Multiply element $a_{31}$ by its cofactor.

$2 | \begin{array}{c} 2 & 4 \\ - 4 & 1 \end{array} |$

Passaggio 6.5.1.9

Add the terms together.

$- 3 (0 + 0 - 1 (3 | \begin{array}{c} - 4 & 1 \\ 3 & 2 \end{array} | + 0 | \begin{array}{c} 2 & 4 \\ 3 & 2 \end{array} | + 2 | \begin{array}{c} 2 & 4 \\ - 4 & 1 \end{array} |) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.2

Moltiplica $0$ per $| \begin{array}{c} 2 & 4 \\ 3 & 2 \end{array} |$ .

$- 3 (0 + 0 - 1 (3 | \begin{array}{c} - 4 & 1 \\ 3 & 2 \end{array} | + 0 + 2 | \begin{array}{c} 2 & 4 \\ - 4 & 1 \end{array} |) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.3

Calcola $| \begin{array}{c} - 4 & 1 \\ 3 & 2 \end{array} |$ .

Tocca per altri passaggi...

Passaggio 6.5.3.1

È possibile trovare il determinante di una matrice $2 \times 2$ usando la formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 3 (0 + 0 - 1 (3 (- 4 \cdot 2 - 3 \cdot 1) + 0 + 2 | \begin{array}{c} 2 & 4 \\ - 4 & 1 \end{array} |) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.3.2

Semplifica il determinante.

Tocca per altri passaggi...

Passaggio 6.5.3.2.1

Semplifica ciascun termine.

Tocca per altri passaggi...

Passaggio 6.5.3.2.1.1

Moltiplica $- 4$ per $2$ .

$- 3 (0 + 0 - 1 (3 (- 8 - 3 \cdot 1) + 0 + 2 | \begin{array}{c} 2 & 4 \\ - 4 & 1 \end{array} |) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.3.2.1.2

Moltiplica $- 3$ per $1$ .

$- 3 (0 + 0 - 1 (3 (- 8 - 3) + 0 + 2 | \begin{array}{c} 2 & 4 \\ - 4 & 1 \end{array} |) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.3.2.2

Sottrai $3$ da $- 8$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 | \begin{array}{c} 2 & 4 \\ - 4 & 1 \end{array} |) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.4

Calcola $| \begin{array}{c} 2 & 4 \\ - 4 & 1 \end{array} |$ .

Tocca per altri passaggi...

Passaggio 6.5.4.1

È possibile trovare il determinante di una matrice $2 \times 2$ usando la formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 (2 \cdot 1 - (- 4 \cdot 4))) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.4.2

Semplifica il determinante.

Tocca per altri passaggi...

Passaggio 6.5.4.2.1

Semplifica ciascun termine.

Tocca per altri passaggi...

Passaggio 6.5.4.2.1.1

Moltiplica $2$ per $1$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 (2 - (- 4 \cdot 4))) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.4.2.1.2

Moltiplica $- (- 4 \cdot 4)$ .

Tocca per altri passaggi...

Passaggio 6.5.4.2.1.2.1

Moltiplica $- 4$ per $4$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 (2 - - 16)) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.4.2.1.2.2

Moltiplica $- 1$ per $- 16$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 (2 + 16)) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.4.2.2

Somma $2$ e $16$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 \cdot 18) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.5

Semplifica il determinante.

Tocca per altri passaggi...

Passaggio 6.5.5.1

Semplifica ciascun termine.

Tocca per altri passaggi...

Passaggio 6.5.5.1.1

Moltiplica $3$ per $- 11$ .

$- 3 (0 + 0 - 1 (- 33 + 0 + 2 \cdot 18) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.5.1.2

Moltiplica $2$ per $18$ .

$- 3 (0 + 0 - 1 (- 33 + 0 + 36) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.5.2

Somma $- 33$ e $0$ .

$- 3 (0 + 0 - 1 (- 33 + 36) + 0) + 0 + 0 + 0 + 0$

Passaggio 6.5.5.3

Somma $- 33$ e $36$ .

$- 3 (0 + 0 - 1 \cdot 3 + 0) + 0 + 0 + 0 + 0$

Passaggio 6.6

Semplifica il determinante.

Tocca per altri passaggi...

Passaggio 6.6.1

Moltiplica $- 1$ per $3$ .

$- 3 (0 + 0 - 3 + 0) + 0 + 0 + 0 + 0$

Passaggio 6.6.2

Somma $0$ e $0$ .

$- 3 (0 - 3 + 0) + 0 + 0 + 0 + 0$

Passaggio 6.6.3

Sottrai $3$ da $0$ .

$- 3 (- 3 + 0) + 0 + 0 + 0 + 0$

Passaggio 6.6.4

Somma $- 3$ e $0$ .

$- 3 \cdot - 3 + 0 + 0 + 0 + 0$

Passaggio 7

Semplifica il determinante.

Tocca per altri passaggi...

Passaggio 7.1

Moltiplica $- 3$ per $- 3$ .

$9 + 0 + 0 + 0 + 0$

Passaggio 7.2

Somma $9$ e $0$ .

$9 + 0 + 0 + 0$

Passaggio 7.3

Somma $9$ e $0$ .

$9 + 0 + 0$

Passaggio 7.4

Somma $9$ e $0$ .

$9 + 0$

Passaggio 7.5

Somma $9$ e $0$ .

$9$