Question

Ajutati un copil sarman ca sa nu ramana corigent, ms <3

Answer 1

Explicație pas cu pas:

a) det(A) =

$\left|\begin{array}{ccc}3&6\\1&2\end{array}\right| = 3 \times 2 - 6 \times 1 = 6 - 6 = 0$

→ det(A) = 0

b)

$A \times A = \left(\begin{array}{ccc}3&6\\1&2\end{array}\right) \times \left(\begin{array}{ccc}3&6\\1&2\end{array}\right) =$

$= \left(\begin{array}{ccc}3 \times 3 + 6 \times 1&3 \times 6 + 6 \times 2\\1 \times 3 + 2 \times 1&1 \times 6 + 2 \times 2\end{array}\right)$

$= \left(\begin{array}{ccc}9 + 6&18 + 12\\3 + 2&6 + 4\end{array}\right)$

$= \left(\begin{array}{ccc}15&30\\5&10\end{array}\right)$

$= \left(\begin{array}{ccc}5 \times 3&5 \times 6\\5 \times 1&5 \times 2\end{array}\right)$

$= 5 \times \left(\begin{array}{ccc}3&6\\1&2\end{array}\right) = 5 \times A$

→ A•A = 5•A

c)

$x \times A + (1 - x)I =$

$= x \times \left(\begin{array}{ccc}3&6\\1&2\end{array}\right) +(1 - x) \left(\begin{array}{ccc}1&0\\0&1\end{array}\right)$

$= \left(\begin{array}{ccc}3x&6x\\x&2x\end{array}\right) + \left(\begin{array}{ccc}1 - x&0\\0&1 - x\end{array}\right)$

$= \left(\begin{array}{ccc}3x + 1 - x&6x\\x&2x + 1 - x\end{array}\right)$

$= \left(\begin{array}{ccc}2x + 1&6x\\x&x + 1\end{array}\right)$

$det\left( x \times A + (1 - x)I\right) =$

$= (2x + 1)(x + 1) - 6x \times x$

$= 2 {x}^{2} + 2x + x + 1 - 6 {x}^{2}$

$= - 4 {x}^{2} + 3x + 1$

$= - (4x + 1)(x - 1)$

$det\left( x \times A + (1 - x)I\right) = 0$

$= > - (4x + 1)(x - 1) = 0$

$4x + 1 = 0 = > x = - \frac{1}{4}$

$x - 1 = 0 = > x = 1$

Answer 2

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