Lineare Algebra Beispiele

[\begin{matrix} \frac{d}{a d - b c} & - \frac{b}{a d - b c} - \frac{c}{a d - b c} & \frac{a}{a d - b d} \end{matrix}] [\begin{matrix} - 6 & k 0 & - 6 \end{matrix}] [\begin{matrix} a & b c & d \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{a d - b d} \end{array}] [\begin{array}{c} - 6 & k \\ 0 & - 6 \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 1

Faktorisiere

d

$d$ aus

a d - b d

$a d - b d$ heraus.

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Schritt 1.1

Faktorisiere

d

$d$ aus

a d

$a d$ heraus.

[\begin{matrix} \frac{d}{a d - b c} & - \frac{b}{a d - b c} - \frac{c}{a d - b c} & \frac{a}{d a - b d} \end{matrix}] [\begin{matrix} - 6 & k 0 & - 6 \end{matrix}] [\begin{matrix} a & b c & d \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{d a - b d} \end{array}] [\begin{array}{c} - 6 & k \\ 0 & - 6 \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 1.2

Faktorisiere

d

$d$ aus

- b d

$- b d$ heraus.

⎡ ⎣ \begin{matrix} \frac{d}{a d - b c} & - \frac{b}{a d - b c} - \frac{c}{a d - b c} & \frac{a}{d a + d (- b)} \end{matrix} ⎤ ⎦ [\begin{matrix} - 6 & k 0 & - 6 \end{matrix}] [\begin{matrix} a & b c & d \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{d a + d (- b)} \end{array}] [\begin{array}{c} - 6 & k \\ 0 & - 6 \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 1.3

Faktorisiere

d

$d$ aus

d a + d (- b)

$d a + d (- b)$ heraus.

⎡ ⎣ \begin{matrix} \frac{d}{a d - b c} & - \frac{b}{a d - b c} - \frac{c}{a d - b c} & \frac{a}{d (a - b)} \end{matrix} ⎤ ⎦ [\begin{matrix} - 6 & k 0 & - 6 \end{matrix}] [\begin{matrix} a & b c & d \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{d (a - b)} \end{array}] [\begin{array}{c} - 6 & k \\ 0 & - 6 \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

⎡ ⎣ \begin{matrix} \frac{d}{a d - b c} & - \frac{b}{a d - b c} - \frac{c}{a d - b c} & \frac{a}{d (a - b)} \end{matrix} ⎤ ⎦ [\begin{matrix} - 6 & k 0 & - 6 \end{matrix}] [\begin{matrix} a & b c & d \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

Schritt 2

Multipliziere

⎡ ⎣ \begin{matrix} \frac{d}{a d - b c} & - \frac{b}{a d - b c} - \frac{c}{a d - b c} & \frac{a}{d (a - b)} \end{matrix} ⎤ ⎦ [\begin{matrix} - 6 & k 0 & - 6 \end{matrix}]

$[\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{d (a - b)} \end{array}] [\begin{array}{c} - 6 & k \\ 0 & - 6 \end{array}]$ .

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Schritt 2.1

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

2 \times 2

$2 \times 2$ and the second matrix is

2 \times 2

$2 \times 2$ .

Schritt 2.2

Multipliziere jede Zeile in der ersten Matrix mit jeder Spalte in der zweiten Matrix.

⎡ ⎣ \begin{matrix} \frac{d}{a d - b c} \cdot - 6 - \frac{b}{a d - b c} \cdot 0 & \frac{d}{a d - b c} k - \frac{b}{a d - b c} \cdot - 6 - \frac{c}{a d - b c} \cdot - 6 + \frac{a}{d (a - b)} \cdot 0 & - \frac{c}{a d - b c} k + \frac{a}{d (a - b)} \cdot - 6 \end{matrix} ⎤ ⎦ [\begin{matrix} a & b c & d \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} \frac{d}{a d - b c} \cdot - 6 - \frac{b}{a d - b c} \cdot 0 & \frac{d}{a d - b c} k - \frac{b}{a d - b c} \cdot - 6 \\ - \frac{c}{a d - b c} \cdot - 6 + \frac{a}{d (a - b)} \cdot 0 & - \frac{c}{a d - b c} k + \frac{a}{d (a - b)} \cdot - 6 \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 2.3

Vereinfache jedes Element der Matrix durch Ausmultiplizieren aller Ausdrücke.

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 d}{a d - b c} & \frac{d k + 6 b}{a d - b c} \frac{6 c}{a d - b c} & - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ [\begin{matrix} a & b c & d \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 d}{a d - b c} & \frac{d k + 6 b}{a d - b c} \\ \frac{6 c}{a d - b c} & - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} \end{array}] [\begin{array}{c} a & b \\ c & d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 d}{a d - b c} & \frac{d k + 6 b}{a d - b c} \frac{6 c}{a d - b c} & - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ [\begin{matrix} a & b c & d \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

Schritt 3

Multipliziere

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 d}{a d - b c} & \frac{d k + 6 b}{a d - b c} \frac{6 c}{a d - b c} & - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ [\begin{matrix} a & b c & d \end{matrix}]

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Schritt 3.1

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

2 \times 2

$2 \times 2$ and the second matrix is

2 \times 2

$2 \times 2$ .

Schritt 3.2

Multipliziere jede Zeile in der ersten Matrix mit jeder Spalte in der zweiten Matrix.

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 d}{a d - b c} a + \frac{d k + 6 b}{a d - b c} c & - \frac{6 d}{a d - b c} b + \frac{d k + 6 b}{a d - b c} d \frac{6 c}{a d - b c} a - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} c & \frac{6 c}{a d - b c} b - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} d \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 d}{a d - b c} a + \frac{d k + 6 b}{a d - b c} c & - \frac{6 d}{a d - b c} b + \frac{d k + 6 b}{a d - b c} d \\ \frac{6 c}{a d - b c} a - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} c & \frac{6 c}{a d - b c} b - \frac{k c d a - k c d b + 6 a^{2} d - 6 a b c}{d (a - b) (a d - b c)} d \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 3.3

Vereinfache jedes Element der Matrix durch Ausmultiplizieren aller Ausdrücke.

Tippen, um mehr Schritte zu sehen ...

Schritt 3.3.1

Wende das Distributivgesetz an.

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a + 6 c b (- b) - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a + 6 c b (- b) - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 3.3.2

Multipliziere

b

$b$ mit

b

$b$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 3.3.2.1

Bewege

b

$b$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a + 6 c (b \cdot b) \cdot - 1 - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a + 6 c (b \cdot b) \cdot - 1 - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 3.3.2.2

Mutltipliziere

b

$b$ mit

b

$b$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a + 6 c b^{2} \cdot - 1 - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a + 6 c b^{2} \cdot - 1 - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a + 6 c b^{2} \cdot - 1 - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

Schritt 3.3.3

Mutltipliziere

- 1

$- 1$ mit

6

$6$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - (k c d a - k c d b + 6 a^{2} d - 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 3.3.4

Wende das Distributivgesetz an.

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - (k c d a) - (- k c d b) - (6 a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - (k c d a) - (- k c d b) - (6 a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 3.3.5

Vereinfache.

Tippen, um mehr Schritte zu sehen ...

Schritt 3.3.5.1

Multipliziere

- (- k c d b)

$- (- k c d b)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 3.3.5.1.1

Mutltipliziere

- 1

$- 1$ mit

- 1

$- 1$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + 1 (k c d b) - (6 a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + 1 (k c d b) - (6 a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 3.3.5.1.2

Mutltipliziere

k

$k$ mit

1

$1$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - (6 a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - (6 a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - (6 a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

Schritt 3.3.5.2

Mutltipliziere

6

$6$ mit

- 1

$- 1$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - 6 (a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - 6 (a^{2} d) - (- 6 a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 3.3.5.3

Mutltipliziere

- 6

$- 6$ mit

- 1

$- 1$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - 6 (a^{2} d) + 6 (a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - 6 (a^{2} d) + 6 (a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - 6 (a^{2} d) + 6 (a b c)}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k (c d b) - 6 (a^{2} d) + 6 (a b c)}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 3.3.6

Entferne die Klammern.

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d + 6 a b c}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 c b a - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d + 6 a b c}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 3.3.7

Addiere

6 c b a

$6 c b a$ und

6 a b c

$6 a b c$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 3.3.7.1

Bewege

b

$b$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 a b c + 6 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{6 a b c + 6 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Schritt 3.3.7.2

Addiere

6 a b c

$6 a b c$ und

6 a b c

$6 a b c$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{12 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} \\ - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{12 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} \end{array}] = [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{12 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{12 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

⎡ ⎢ ⎣ \begin{matrix} - \frac{6 a d - c d k - 6 c b}{a d - b c} & \frac{d^{2} k}{a d - b c} - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} & \frac{12 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

Schritt 4

Write as a linear system of equations.

$- \frac{6 a d - c d k - 6 c b}{a d - b c} = 1$

$\frac{d^{2} k}{a d - b c} = 0$

$- \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} = 0$

$\frac{12 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} = 1$

Schritt 5

Vereinfache das System.

$- \frac{6 a d - c d k - 6 c b}{a d - b c} = 1, \frac{d^{2} k}{a d - b c} = 0, - \frac{c (6 a d b + k c d a - k c d b - 6 a b c)}{d (a - b) (a d - b c)} = 0, \frac{12 a b c - 6 c b^{2} - k c d a + k c d b - 6 a^{2} d}{(a - b) (a d - b c)} = 1$