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If three numbers are added their sum is 2.1f 2 times the second number is subtracted from the sum of first and third number we get 8 and if three times the first number is added to the sum of second and third number we get 4. Find the numbers using matrix method. |
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Answer» Let numbers are x, y and z x + y + z = 2 x + z - 2y = 8 3x + y + z = 4 Matrix form is \(\begin{bmatrix}1&1&1\\1&1& -2\\3& 1&1\end{bmatrix}\)\(\begin{bmatrix} x\\y\\z\end{bmatrix}\) = \(\begin{bmatrix} 2\\8\\4\end{bmatrix}\) ∴ \(\begin{bmatrix} x\\y\\z\end{bmatrix}\) = \(\begin{bmatrix}1&1&1\\1&1& -2\\3& 1&1\end{bmatrix}^{-1}\)\(\begin{bmatrix} 2\\8\\4\end{bmatrix}\) Let A = \(\begin{bmatrix}1&1&1\\1&1& -2\\3& 1&1\end{bmatrix}\) A = IA \(\begin{bmatrix}1&1&1\\1&1& -2\\3& 1&1\end{bmatrix}\) = \(\begin{bmatrix}1&0&0\\0&1& 0\\0& 0&1\end{bmatrix}\)A Applying R2→R2 - R1, R3 → R3 - 3R1 \(\begin{bmatrix}1&1&1\\0&0& -3\\0& -2&-2\end{bmatrix}\) = \(\begin{bmatrix}1&0&0\\-1&1& 0\\-3& 0&1\end{bmatrix}A\) Applying R3→R2 \(\begin{bmatrix}1&1&1\\0&-2& -2\\0& 0&-3\end{bmatrix}\) = \(\begin{bmatrix}1&0&0\\-3&0& 1\\-1& 1&0\end{bmatrix}A\) Applying R1→R2/-2, R3 → R3/-3 \(\begin{bmatrix}1&1&1\\0&1& 1\\0& 0&1\end{bmatrix}\) = \(\begin{bmatrix}1&0&0\\3/2&0& -1/2\\1/3& -1/3&0\end{bmatrix}A\) \(\begin{bmatrix}1&0&0\\0&1& 0\\0& 0&1\end{bmatrix}\) = \(\begin{bmatrix}-1/2&0&1/2\\7/6&1/3& -1/2\\1/3& -1/3&0\end{bmatrix}\) ∴ A-1 = \(\begin{bmatrix}-1/2&0&1/2\\7/6&1/3& -1/2\\1/3& -1/3&0\end{bmatrix}\) ∴ x = A-1b = \(\begin{bmatrix}-1/2&0&1/2\\7/6&1/3& -1/2\\1/3& -1/3&0\end{bmatrix}\)\(\begin{bmatrix} 2\\8\\4\end{bmatrix}\) \(\begin{bmatrix} x\\y\\z\end{bmatrix}\) = \(\begin{bmatrix} 1\\3\\-2\end{bmatrix}\) ∴ Numbers are 1, 3 and -2 |
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