Question

Construct a 3x3 nonzero matrix A such that the vector [1, 0, -1]^T is a solution of Ax = 0.

a) [1, 0, -1; 0, 1, 0; -1, 0, 1]
b) [0, 1, 0; 1, 0, -1; -1, 0, 1]
c) [1, 0, -1; 0, -1, 0; -1, 0, 1]
d) [1, 0, -1; 0, 1, 0; 0, 0, 0]

Answer 1

To construct a 3x3 nonzero matrix A such that the vector [1, 0, -1]^T is a solution of Ax = 0, you need to find a matrix that when multiplied by the given vector, results in the zero vector. The correct answer is option c) [1, 0, -1; 0, -1, 0; -1, 0, 1].

Step-by-step explanation:

To construct a 3x3 nonzero matrix A such that the vector [1, 0, -1]^T is a solution of Ax = 0, we need to find a matrix that when multiplied by the given vector, results in the zero vector.

The correct answer is option c) [1, 0, -1; 0, -1, 0; -1, 0, 1].

Multiplying this matrix by the vector [1, 0, -1]^T:

[1*1 + 0*0 + (-1)*(-1), 1*0 + 0*(-1) + (-1)*0, 1*(-1) + 0*0 + (-1)*1] = [1 + 0 + 1, 0 + 0, -1 - 0 -1] = [2, 0, -2].

Therefore, the matrix A = [1, 0, -1; 0, -1, 0; -1, 0, 1] when multiplied by the vector [1, 0, -1]^T results in the zero vector.