Matrix transformation: Let f: R^3 → R^3 be the matrix transformation defined by

Question

Answer 1

\(\begin{bmatrix}1&2&3\\-3&-2&-1\\-2&0&2\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}2\\2\\4\end{bmatrix} \)

⇒ \(\begin{bmatrix}x+2y+3z\\-3x-2y-z\\-2x+2z\end{bmatrix} = \begin{bmatrix}2\\2\\4\end{bmatrix} \)

⇒ \(-2x + 2z = 4\)

⇒ \(-x + z = 2\)

Also,

\(x + 2y + 3z = 2\)

⇒ \(x + 2y + 3(2 + x) =2\)

⇒ \(4x + 2y = -4\)

⇒ \(2x + y = -2\)

And

\(-3x - 2y - z = 2\)

⇒ \(-3x - 2(-2 - 2x) - (2 + x) = 2\)

⇒ \(-3x + 4x + 4-2-x = 2\)

⇒ \(2 = 2\)  (True)

So, infinite many solution.

Let

Hence, [x, -2 -2x, 2 + x]^T is solution of given system.

There are infinitely many solutions depends on the different values of x.

If x = 0 then y = -2 & z = 2 means [0, -2, 2]^T is a solution of given system.