Given the matrix \( A = \begin{bmatrix} 1 & 2 & -1\ 2 & 4 & -2\ -1 & -2 & 1 \end{bmatrix} \), find the nullity of \( A \).
Answer
Step 3: Solving this system, we see that the system is dependent and the solution is a line in \( R^3 \), which is a one-dimensional space. Therefore, the nullity of \( A \) is 1.
Step 1 ：Step 1: The nullity of a matrix is the dimension of the null space of the matrix. The null space of a matrix \( A \) is the set of all vectors \( x \) such that \( Ax = 0 \).
Step 2 ：Step 2: To find the null space of the matrix, we solve the system \( Ax = 0 \). In this case, this gives us the system \( \begin{bmatrix} 1 & 2 & -1\ 2 & 4 & -2\ -1 & -2 & 1 \end{bmatrix} \begin{bmatrix} x1\ x2\ x3 \end{bmatrix} = 0 \), which simplifies to \( x1+2x2-x3=0, 2x1+4x2-2x3=0, -x1-2x2+x3=0 \).
Step 3 ：Step 3: Solving this system, we see that the system is dependent and the solution is a line in \( R^3 \), which is a one-dimensional space. Therefore, the nullity of \( A \) is 1.
