Aljabar Linear Contoh

$[\begin{array}{c} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}] y = [\begin{array}{c} 1 & 2 \\ 3 & 3 \\ 2 & 1 \end{array}]$

Langkah 1

Find the inverse of $[\begin{array}{c} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}]$ .

Ketuk untuk lebih banyak langkah...

Langkah 1.1

Tulis kembali.

$| \begin{array}{c} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array} |$

Langkah 1.2

Find the determinant.

Ketuk untuk lebih banyak langkah...

Langkah 1.2.1

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 $1$ by its cofactor and add.

Ketuk untuk lebih banyak langkah...

Langkah 1.2.1.1

Consider the corresponding sign chart.

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

Langkah 1.2.1.2

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

Langkah 1.2.1.3

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

$| \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} |$

Langkah 1.2.1.4

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

$1 | \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} |$

Langkah 1.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} |$

Langkah 1.2.1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} |$

Langkah 1.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Langkah 1.2.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Langkah 1.2.1.9

Add the terms together.

$1 | \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Langkah 1.2.2

Kalikan $0$ dengan $| \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} |$ .

$1 | \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} | + 0 + 0 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Langkah 1.2.3

Kalikan $0$ dengan $| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$ .

$1 | \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} | + 0 + 0$

Langkah 1.2.4

Evaluasi $| \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} |$ .

Ketuk untuk lebih banyak langkah...

Langkah 1.2.4.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

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

Langkah 1.2.4.2

Sederhanakan determinannya.

Ketuk untuk lebih banyak langkah...

Langkah 1.2.4.2.1

Kalikan $1$ dengan $1$ .

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

Langkah 1.2.4.2.2

Kurangi $0$ dengan $1$ .

$1 \cdot 1 + 0 + 0$

Langkah 1.2.5

Sederhanakan determinannya.

Ketuk untuk lebih banyak langkah...

Langkah 1.2.5.1

Kalikan $1$ dengan $1$ .

$1 + 0 + 0$

Langkah 1.2.5.2

Tambahkan $1$ dan $0$ .

$1 + 0$

Langkah 1.2.5.3

Tambahkan $1$ dan $0$ .

$1$

Langkah 1.3

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

Langkah 1.4

Set up a $3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

$[\begin{array}{c} 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{array}]$

Langkah 1.5

Tentukan bentuk eselon baris yang dikurangi.

Ketuk untuk lebih banyak langkah...

Langkah 1.5.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

Ketuk untuk lebih banyak langkah...

Langkah 1.5.1.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 0 & 0 \\ 1 - 1 & 1 - 0 & 0 - 0 & 0 - 1 & 1 - 0 & 0 - 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{array}]$

Langkah 1.5.1.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & - 1 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{array}]$

Langkah 1.5.2

Perform the row operation $R_{3} = R_{3} - R_{1}$ to make the entry at $3, 1$ a $0$ .

Ketuk untuk lebih banyak langkah...

Langkah 1.5.2.1

Perform the row operation $R_{3} = R_{3} - R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & - 1 & 1 & 0 \\ 1 - 1 & 1 - 0 & 1 - 0 & 0 - 1 & 0 - 0 & 1 - 0 \end{array}]$

Langkah 1.5.2.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & - 1 & 1 & 0 \\ 0 & 1 & 1 & - 1 & 0 & 1 \end{array}]$

Langkah 1.5.3

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

Ketuk untuk lebih banyak langkah...

Langkah 1.5.3.1

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & - 1 & 1 & 0 \\ 0 - 0 & 1 - 1 & 1 - 0 & - 1 + 1 & 0 - 1 & 1 - 0 \end{array}]$

Langkah 1.5.3.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & - 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & - 1 & 1 \end{array}]$

Langkah 1.6

The right half of the reduced row echelon form is the inverse.

$[\begin{array}{c} 1 & 0 & 0 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{array}]$

Langkah 2

Multiply both sides by the inverse of $[\begin{array}{c} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}]$ .

$[\begin{array}{c} 1 & 0 & 0 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}] y = [\begin{array}{c} 1 & 0 & 0 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{array}] [\begin{array}{c} 1 & 2 \\ 3 & 3 \\ 2 & 1 \end{array}]$

Langkah 3

Sederhanakan persamaannya.

Ketuk untuk lebih banyak langkah...

Langkah 3.1

Kalikan $[\begin{array}{c} 1 & 0 & 0 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}]$ .

Ketuk untuk lebih banyak langkah...

Langkah 3.1.1

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 $3 \times 3$ and the second matrix is $3 \times 3$ .

Langkah 3.1.2

Kalikan setiap baris pada matriks pertama dengan setiap kolom pada matriks kedua.

$[\begin{array}{c} 1 \cdot 1 + 0 \cdot 1 + 0 \cdot 1 & 1 \cdot 0 + 0 \cdot 1 + 0 \cdot 1 & 1 \cdot 0 + 0 \cdot 0 + 0 \cdot 1 \\ - 1 \cdot 1 + 1 \cdot 1 + 0 \cdot 1 & - 0 + 1 \cdot 1 + 0 \cdot 1 & - 0 + 1 \cdot 0 + 0 \cdot 1 \\ 0 \cdot 1 - 1 \cdot 1 + 1 \cdot 1 & 0 \cdot 0 - 1 \cdot 1 + 1 \cdot 1 & 0 \cdot 0 - 0 + 1 \cdot 1 \end{array}] y = [\begin{array}{c} 1 & 0 & 0 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{array}] [\begin{array}{c} 1 & 2 \\ 3 & 3 \\ 2 & 1 \end{array}]$

Langkah 3.1.3

Sederhanakan setiap elemen dalam matriks dengan mengalikan semua pernyataannya.

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] y = [\begin{array}{c} 1 & 0 & 0 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{array}] [\begin{array}{c} 1 & 2 \\ 3 & 3 \\ 2 & 1 \end{array}]$

Langkah 3.2

Multiplying the identity matrix by any matrix $A$ is the matrix $A$ itself.

$y = [\begin{array}{c} 1 & 0 & 0 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{array}] [\begin{array}{c} 1 & 2 \\ 3 & 3 \\ 2 & 1 \end{array}]$

Langkah 3.3

Kalikan $[\begin{array}{c} 1 & 0 & 0 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{array}] [\begin{array}{c} 1 & 2 \\ 3 & 3 \\ 2 & 1 \end{array}]$ .

Ketuk untuk lebih banyak langkah...

Langkah 3.3.1

Langkah 3.3.2

Kalikan setiap baris pada matriks pertama dengan setiap kolom pada matriks kedua.

$y = [\begin{array}{c} 1 \cdot 1 + 0 \cdot 3 + 0 \cdot 2 & 1 \cdot 2 + 0 \cdot 3 + 0 \cdot 1 \\ - 1 \cdot 1 + 1 \cdot 3 + 0 \cdot 2 & - 1 \cdot 2 + 1 \cdot 3 + 0 \cdot 1 \\ 0 \cdot 1 - 1 \cdot 3 + 1 \cdot 2 & 0 \cdot 2 - 1 \cdot 3 + 1 \cdot 1 \end{array}]$

Langkah 3.3.3

Sederhanakan setiap elemen dalam matriks dengan mengalikan semua pernyataannya.

$y = [\begin{array}{c} 1 & 2 \\ 2 & 1 \\ - 1 & - 2 \end{array}]$