# Matrix Null Space (Kernel) and Nullity Calculator

The calculator will find the null space (kernel) and the nullity of the given matrix, with steps shown.

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Find the null space of $$\left[\begin{array}{ccc}1 & -1 & -1\\2 & -2 & 1\end{array}\right]$$$. ## Solution The reduced row echelon form of the matrix is $$\left[\begin{array}{ccc}1 & -1 & 0\\0 & 0 & 1\end{array}\right]$$$ (for steps, see rref calculator).
To find the null space, solve the matrix equation $$\left[\begin{array}{ccc}1 & -1 & 0\\0 & 0 & 1\end{array}\right] \left[\begin{array}{c}x_{1}\\x_{2}\\x_{3}\end{array}\right] = \left[\begin{array}{c}0\\0\end{array}\right].$$$If we take $$x_{2} = t$$$, then $$x_{1} = t$$$, $$x_{3} = 0$$$.
Thus, $$\mathbf{\vec{x}} = \left[\begin{array}{c}t\\t\\0\end{array}\right] = \left[\begin{array}{c}1\\1\\0\end{array}\right] t.$$$This is the null space. The nullity of a matrix is the dimension of the basis for the null space. Thus, the nullity of the matrix is $$1$$$.
The basis for the null space is $$\left\{\left[\begin{array}{c}1\\1\\0\end{array}\right]\right\}$$$A. The nullity of the matrix is $$1$$$A.