Question

A= 3 -2
2 1
B= 4 -3
3 1
I₂ = 1 0
0 1


Determinati matricea x€ M₂ (R) pentru care A•X-B•X = I₂-X
A•X - B•X = I₂ - X => ( A-B +I₂) • X = I₂ ​

Answer 1

Explicație pas cu pas:

$A = \left[\begin{array}{ccc}3&-2\\2&1\end{array}\right]$

$B = \left[\begin{array}{ccc}4&-3\\3&1\end{array}\right]$

$I_{2} = \left[\begin{array}{ccc}1&0\\0&1\end{array}\right]$

notăm:

$X = \left[\begin{array}{ccc}a&b\\c&d\end{array}\right]$

→

$A\cdot X = \left[\begin{array}{ccc}3&-2\\2&1\end{array}\right]\cdot \left[\begin{array}{ccc}a&b\\c&d\end{array}\right] \\ = \left[\begin{array}{ccc}3a-2c&3b-2d\\2a+c&2b+d\end{array}\right]$

$B\cdot X = \left[\begin{array}{ccc}4&-3\\3&1\end{array}\right]\cdot \left[\begin{array}{ccc}a&b\\c&d\end{array}\right]\\ = \left[\begin{array}{ccc}4a-3c&4b-3d\\3a+c&3b+d\end{array}\right]$

$I_{2} - X = \left[\begin{array}{ccc}1&0\\0&1\end{array}\right] - \left[\begin{array}{ccc}a&b\\c&d\end{array}\right] \\ = \left[\begin{array}{ccc}1-a&-b\\-c&1-d\end{array}\right]$

$A\cdot X - B\cdot X = \\ = \left[\begin{array}{ccc}3a-2c&3b-2d\\2a+c&2b+d\end{array}\right] - \left[\begin{array}{ccc}4a-3c&4b-3d\\3a+c&3b+d\end{array}\right] \\ = \left[\begin{array}{ccc}3a-2c-4a+3c&3b-2d-4b+3d\\2a+c-3a-c&2b+d-3b-d\end{array}\right] \\ = \left[\begin{array}{ccc}c-a&d-b\\-a&-b\end{array}\right]$

$A\cdot X - B\cdot X = I_{2} - X \\ \left[\begin{array}{ccc}c-a&d-b\\-a&-b\end{array}\right] = \left[\begin{array}{ccc}1-a&-b\\-c&1-d\end{array}\right]$

$\left\{\begin{array}{c}c-a=1-a \\ d-b=-b \\ -a=-c \\ -b=1-d \end{array}\right. =>\left\{\begin{array}{c}c=1 \\ d=0 \\ a=1 \\ b=-1 \end{array}\right.$

$=> X = \left[\begin{array}{ccc}1&-1\\1&0\end{array}\right]$