Question

$A= \left[\begin{array}{ccc}2&0\\-1&3\\\end{array}\right] B= \left[\begin{array}{ccc}0&1\\5&-1\\\end{array}\right] C= \left[\begin{array}{ccc}4&5\\2&1\\\end{array}\right]$

a matriz X tal que 2x+b^t=A - C vale?

Answer 1

$X=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\\\\2X=2\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\\\\2X=\left[\begin{array}{cc}2a&2b\\2c&2d\end{array}\right]$
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$B=\left[\begin{array}{cc}0&1\\5&-1\end{array}\right]\\\\B^{T}=\left[\begin{array}{cc}0&5\\1&-1\end{array}\right]$

$A - C=\left[\begin{array}{cc}2&0\\-1&3\end{array}\right]-\left[\begin{array}{cc}4&5\\2&1\end{array}\right]=\left[\begin{array}{cc}(2-4)&(0-5)\\(-1-2)&(3-1)\end{array}\right]$

$A-C=\left[\begin{array}{cc}-2&-5\\-3&2\end{array}\right]$
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$2X+B^{T}=A-C\\\\ \left[\begin{array}{cc}2a&2b\\2c&2d\end{array}\right]+\left[\begin{array}{cc}0&5\\1&-1\end{array}\right]=\left[\begin{array}{cc}-2&-5\\-3&2\end{array}\right]\\\\\left[\begin{array}{cc}2a+0&2b+5\\2c+1&2d-1\end{array}\right]=\left[\begin{array}{cc}-2&-5\\-3&2\end{array}\right]\\\\\\2a+0=-2\\2a=-2\\a=-2/2\\a=-1$

$2b+5=-5\\2b=-5-5\\2b=-10\\b=-10/2\\b=-5$

$2c+1=-3\\2c=-3-1\\2c=-4\\c=-4/2\\c=-2\\\\2d-1=2\\2d=2+1\\2d=3\\d=3/2$
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$X= \left[\begin{array}{cc}a&b\\c&d\end{array}\right] \\\\X= \left[\begin{array}{cc}-1&-5\\-2&\frac{3}{2}\end{array}\right]$