Question

Solve the given equation using matrix method x+y+z=3 2x+3y+4z=12 x-y-2z =-5​

Answer 1

Answer:

x = 1

y = -2

z = 4

Step-by-step explanation:

Given:

• x + y + z = 3
• 2x + 3y + 4z = 12
• x - y - 2z = -5

To Find:

• The solution of the given equations using matrix method,

Solution:

$\sf Let\:A=\left[\begin{array}{ccc}1&1&1\\2&3&4\\1&-1&-2\end{array}\right] , X=\left[\begin{array}{c}x\\y\\z\end{array}\right], B=\left[\begin{array}{c}3\\12\\-5\end{array}\right]$

This can be written as AX = B

Hence,

X = A⁻¹ B

Finding the inverse of matrix A,

$\sf A^{-1}=\dfrac{adj\:A}{|A|}$

|A| = 1 ( -6 + 4) -1 (-4 - 4) + 1( -2 - 3)

= -2 + 8 -5

= 1

$\sf adj\:A=\left[\begin{array}{ccc}-2&8&-5\\1&-3&2\\1&-2&1\end{array}\right] ^T$

$\sf adj\:A=\left[\begin{array}{ccc}-2&1&1\\8&-3&-2\\-5&2&1\end{array}\right]$

Hence,

$\sf A^{-1}=\left[\begin{array}{ccc}-2&1&1\\8&-3&-2\\-5&2&1\end{array}\right]$

Now we know that,

X = A⁻¹ B

Therefore,

$\sf \left[\begin{array}{c}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}-2&1&1\\8&-3&-2\\-5&2&1\end{array}\right] \: \left[\begin{array}{c}3\\12\\-5\end{array}\right]$

$\sf \left[\begin{array}{c}x\\y\\z\end{array}\right] =\left[\begin{array}{c}-6+12-5\\24-36+10\\-15+24-5\end{array}\right]$

$\sf \left[\begin{array}{c}x\\y\\z\end{array}\right] =\left[\begin{array}{c}1\\-2\\4\end{array}\right]$

Therefore x = 1, y = -2 and z = 4