Aljabar Linear Contoh

[\begin{matrix} a & b c & d \end{matrix}]

$[\begin{array}{c} a & b \\ c & d \end{array}]$

Langkah 1

The inverse of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

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

$\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where

a d - b c

$a d - b c$ is the determinant.

Langkah 2

Determinan dari matriks

2 \times 2

$2 \times 2$ dapat dicari menggunakan rumus

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a d - c b

$a d - c b$

Langkah 3

Since the determinant is non-zero, the inverse exists.

Langkah 4

Substitute the known values into the formula for the inverse.

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

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

Langkah 5

Kalikan

\frac{1}{a d - c b}

$\frac{1}{a d - c b}$ dengan setiap elemen di dalam matriks tersebut.

[\begin{matrix} \frac{1}{a d - c b} d & \frac{1}{a d - c b} (- b) \frac{1}{a d - c b} (- c) & \frac{1}{a d - c b} a \end{matrix}]

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

Langkah 6

Sederhanakan setiap elemen dalam matriks.

Ketuk untuk lebih banyak langkah...

Langkah 6.1

Gabungkan

\frac{1}{a d - c b}

$\frac{1}{a d - c b}$ dan

d

$d$ .

[\begin{matrix} \frac{d}{a d - c b} & \frac{1}{a d - c b} (- b) \frac{1}{a d - c b} (- c) & \frac{1}{a d - c b} a \end{matrix}]

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

Langkah 6.2

Tulis kembali menggunakan sifat komutatif dari perkalian.

[\begin{matrix} \frac{d}{a d - c b} & - \frac{1}{a d - c b} b \frac{1}{a d - c b} (- c) & \frac{1}{a d - c b} a \end{matrix}]

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

Langkah 6.3

Gabungkan

b

$b$ dan

\frac{1}{a d - c b}

$\frac{1}{a d - c b}$ .

[\begin{matrix} \frac{d}{a d - c b} & - \frac{b}{a d - c b} \frac{1}{a d - c b} (- c) & \frac{1}{a d - c b} a \end{matrix}]

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

Langkah 6.4

Tulis kembali menggunakan sifat komutatif dari perkalian.

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

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

Langkah 6.5

Gabungkan

c

$c$ dan

\frac{1}{a d - c b}

$\frac{1}{a d - c b}$ .

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

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

Langkah 6.6

Gabungkan

\frac{1}{a d - c b}

$\frac{1}{a d - c b}$ dan

a

$a$ .

[\begin{matrix} \frac{d}{a d - c b} & - \frac{b}{a d - c b} - \frac{c}{a d - c b} & \frac{a}{a d - c b} \end{matrix}]

$[\begin{array}{c} \frac{d}{a d - c b} & - \frac{b}{a d - c b} \\ - \frac{c}{a d - c b} & \frac{a}{a d - c b} \end{array}]$

[\begin{matrix} \frac{d}{a d - c b} & - \frac{b}{a d - c b} - \frac{c}{a d - c b} & \frac{a}{a d - c b} \end{matrix}]

$[\begin{array}{c} \frac{d}{a d - c b} & - \frac{b}{a d - c b} \\ - \frac{c}{a d - c b} & \frac{a}{a d - c b} \end{array}]$