The value of the matrix X is [tex]X = \left[\begin{array}{cc}0.5&0\\1&-0.5\end{array}\right][/tex]
How to solve for matrix X?
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The given matrices are:
[tex]A = \left[\begin{array}{cc}5&-3\\4&-2\end{array}\right][/tex]
[tex]B = \left[\begin{array}{cc}-1&1\\0&2\end{array}\right][/tex]
[tex]C = \left[\begin{array}{cc}-1&2\\-2&2\end{array}\right][/tex]
The expression is given as:
A(X - B) = C
Expand
AX - BX = C
This gives
[tex]\left[\begin{array}{cc}5&-3\\4&-2\end{array}\right] * \left[\begin{array}{cc}a&b\\c&d\end{array}\right] - \left[\begin{array}{cc}-1&1\\0&2\end{array}\right] * \left[\begin{array}{cc}a&b\\c&d\end{array}\right] = \left[\begin{array}{cc}-1&2\\-2&2\end{array}\right][/tex]
Evaluate the products
[tex]\left[\begin{array}{cc}5a - 3c&5b -3d\\4a - 2c&4b -2d\end{array}\right] - \left[\begin{array}{cc}-a + c&-b + d\\2c&2d\end{array}\right] = \left[\begin{array}{cc}-1&2\\-2&2\end{array}\right][/tex]
Evaluate the difference
[tex]\left[\begin{array}{cc}6a - 4c&6b -4d\\4a - 4c&4b -4d\end{array}\right] = \left[\begin{array}{cc}-1&2\\-2&2\end{array}\right][/tex]
By comparing the positions, we have:
6a - 4c = -1
6b - 4d = 2
4a - 4c = -2
4b - 4d = 2
Rewrite as:
6a - 4c = -1 and 4a - 4c = -2
6b - 4d = 2 and 4b - 4d = 2
Using a graphing tool, we have:
a = 0.5, c = 1, b = 0, and d = -0.5
Hence, the matrix X is:
[tex]X = \left[\begin{array}{cc}0.5&0\\1&-0.5\end{array}\right][/tex]
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