Álgebra linear Exemplos

⎡ ⎢ ⎣ \begin{matrix} 0 & cos (x) & sin (x) 0 & - 2 cos (x) - sin (x) & cos (x) - 2 sin (x) 0 & 3 cos (x) + 4 sin (x) & - 4 cos (x) + 3 sin (x) \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 0 & \cos (x) & \sin (x) \\ 0 & - 2 \cos (x) - \sin (x) & \cos (x) - 2 \sin (x) \\ 0 & 3 \cos (x) + 4 \sin (x) & - 4 \cos (x) + 3 \sin (x) \end{array}]$

Step 1

Multiply every element in the

1 s t

$1 s t$ column by its cofactor and add.

0 ∣ ∣ ∣ \begin{matrix} - 2 cos (x) - sin (x) & cos (x) - 2 sin (x) 3 cos (x) + 4 sin (x) & - 4 cos (x) + 3 sin (x) \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} cos (x) & sin (x) 3 cos (x) + 4 sin (x) & - 4 cos (x) + 3 sin (x) \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} cos (x) & sin (x) - 2 cos (x) - sin (x) & cos (x) - 2 sin (x) \end{matrix} ∣ ∣ ∣

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

Step 2

Multiplique

0

$0$ por

∣ ∣ ∣ \begin{matrix} - 2 cos (x) - sin (x) & cos (x) - 2 sin (x) 3 cos (x) + 4 sin (x) & - 4 cos (x) + 3 sin (x) \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} - 2 \cos (x) - \sin (x) & \cos (x) - 2 \sin (x) \\ 3 \cos (x) + 4 \sin (x) & - 4 \cos (x) + 3 \sin (x) \end{array} |$ .

0 + 0 ∣ ∣ ∣ \begin{matrix} cos (x) & sin (x) 3 cos (x) + 4 sin (x) & - 4 cos (x) + 3 sin (x) \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} cos (x) & sin (x) - 2 cos (x) - sin (x) & cos (x) - 2 sin (x) \end{matrix} ∣ ∣ ∣

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

Step 3

Multiplique

0

$0$ por

∣ ∣ ∣ \begin{matrix} cos (x) & sin (x) 3 cos (x) + 4 sin (x) & - 4 cos (x) + 3 sin (x) \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} \cos (x) & \sin (x) \\ 3 \cos (x) + 4 \sin (x) & - 4 \cos (x) + 3 \sin (x) \end{array} |$ .

0 + 0 + 0 ∣ ∣ ∣ \begin{matrix} cos (x) & sin (x) - 2 cos (x) - sin (x) & cos (x) - 2 sin (x) \end{matrix} ∣ ∣ ∣

$0 + 0 + 0 | \begin{array}{c} \cos (x) & \sin (x) \\ - 2 \cos (x) - \sin (x) & \cos (x) - 2 \sin (x) \end{array} |$

Step 4

Multiplique

0

$0$ por

∣ ∣ ∣ \begin{matrix} cos (x) & sin (x) - 2 cos (x) - sin (x) & cos (x) - 2 sin (x) \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} \cos (x) & \sin (x) \\ - 2 \cos (x) - \sin (x) & \cos (x) - 2 \sin (x) \end{array} |$ .

0 + 0 + 0

$0 + 0 + 0$

Step 5

Simplifique o determinante.

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Some

0

$0$ e

0

$0$ .

0 + 0

$0 + 0$

Some

0

$0$ e

0

$0$ .

0

$0$

0

$0$