Question

श्न-27
आव्यूह विधि से निम्न समीकरण निकाय को हल कीजिए:
x+ y + z=3
2x-y+z=2
x-2y+ 32 = 2
Solve the following system of equations by matrix method:
x+y+z=3
2x-y+z=2
x-2y+32 = 2​

Answer 1

Step-by-step explanation:

Given:

$x+y+z=3\\ 2x-y+z=2\\ x-2y+3z=2$

To find:Solution of systems of equations by matrix method.

Solution:

Step 1:To find the solution by matrix method, write the equation in terms of matrix

AX=B

$\left[\begin{array}{ccc}1&1&1\\2&-1&1\\1&-2&3\end{array}\right] \left[\begin{array}{c}x\\y\\z\end{array}\right] =\left[\begin{array}{c}3\\2\\2\end{array}\right]$

$X=A^{-1}B$

Step 2: Find inverse of A

For that write Adj A and find determinant of A

$|A|=\left|\begin{array}{ccc}1&1&1\\2&-1&1\\1&-2&3\end{array}\right|\\ \\ |A|=1(-3+2) -1(6-1)+1(-4+1)\\ \\ |A|=-9$

Write Adj.A

$Adj.A=\left[\begin{array}{ccc}-1&-5&2\\-5&2&1\\-3&3&-3\end{array}\right] \\ \\ \\ A^{-1}=\frac{-1}{9}\left[\begin{array}{ccc}-1&-5&2\\-5&2&1\\-3&3&-3\end{array}\right]\\ \\ or\\ \\ A^{-1}=\frac{1}{9}\left[\begin{array}{ccc}1&5&-2\\5&-2&-1\\3&-3&3\end{array}\right]$

Step 3: Find the unknown matrix by writing the values into

$X=A^{-1} \times B\\ \\ \left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\frac{1}{9}\left[\begin{array}{ccc}1&5&-2\\5&-2&-1\\3&-3&3\end{array}\right] \left[\begin{array}{ccc}3\\2\\2\end{array}\right]$

Step 4: Multiply matrix to find the value of x,y and z

$\left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\frac{1}{9} \left[\begin{array}{ccc}3\times1+2\times 5+2\times(-2)\\3\times5+2\times (-2)+2\times(-1)\\3\times3+2\times(-3)+2\times 3\end{array}\right] \\\\ \\ \left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\frac{1}{9}\left[\begin{array}{ccc}9\\9\\9\end{array}\right]$

$\left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}1\\1\\1\end{array}\right]$

Final answer:

$x=1\\ y=1\\ z=1$

Hope it helps you.

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