# Consider matrix A[[2,1,1],[-2,3,-1]]. find the space N(A). describe it geometrically

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sciencesolve | Certified Educator

You need to find the nul space of matrix A, hence, you need to solve the equation, such that:

`A*x = 0`

`((2,1,1),(-2,3,-1))*((x_1),(x_2),(x_3)) = ((0),(0))`

`((2x_1 + x_2 + x_3),(-2x_1 + 3x_2 - x_3)) = ((0),(0))`

Equating the corresponding members yields:

`{(2x_1 + x_2 + x_3 = 0),(-2x_1 + 3x_2 - x_3 = 0):} `

You may consider variable x_3 as free variable and move it to the right, such that:

`{(2x_1 + x_2 = - x_3),(-2x_1 + 3x_2 = x_3):} `

Adding the equations yields:

`4x_2 = 0 => x_2 = 0`

Replacing 0 for `x_2` in equation `2x_1 + x_2 = - x_3` yields:

`2x_1 + 0 = - x_3 => x_1 = -x_3/2`

**Hence, evaluating the null space of matrix A, yields **`N(A) = ((-x_3/2),(0),(x_3)).`