Question

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.

Answer 1

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

\(\begin{bmatrix}1&1&1\\0&1& 1\\0& 0&1\end{bmatrix}\) = \(\begin{bmatrix}1&0&0\\3/2&0& -1/2\\1/3& -1/3&0\end{bmatrix}A\)
Applying R₁ → R₁ - R₂, R₂ → R₂ - R₃

\(\begin{bmatrix}1&0&0\\0&1& 0\\0& 0&1\end{bmatrix}\) = \(\begin{bmatrix}-1/2&0&1/2\\7/6&1/3& -1/2\\1/3& -1/3&0\end{bmatrix}\)

∴ 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