Given the matrix ( A = begin{bmatrix} 1 & 2 & -1 2 & 4 & -2 -1 & -2 & 1 end{bmatrix} ), find the nullity of ( A ).

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Accepted Answer

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.

Answer

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.

Given the matrix $A = [\begin{matrix} 1 & 2 & - 1 2 & 4 & - 2 - 1 & - 2 & 1 \end{matrix}]$ , find the nullity of $A$ .

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Given the matrix A=[12−1 24−2 −1−21], find the nullity of A.

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Popular Problems

Given the matrix $A = [\begin{matrix} 1 & 2 & - 1 2 & 4 & - 2 - 1 & - 2 & 1 \end{matrix}]$ , find the nullity of $A$ .