Questions: Consider the following matrices. A = [ [3, -1, -1], [1, -3, 3] ] x = [ [x1], [x2], [x3] ] 0 = [ [0], [0] ] Solve the matrix equation Ax=0. (If there is no solution, enter No Solution. If the system has an infinite number of solutions, express x1, x2, and x3 in terms of the parameter t.) (x1, x2, x3) = ( )

Solution Steps

To solve the matrix equation \( A \vec{x} = \overrightarrow{0} \), we need to find the null space of matrix \( A \). This involves performing row reduction (Gaussian elimination) on the augmented matrix \([A | \overrightarrow{0}]\) to find the solutions for \(\vec{x}\). If the system has free variables, it indicates an infinite number of solutions, which can be expressed in terms of a parameter.

We are given the matrix equation \( A \vec{x} = \overrightarrow{0} \), where \[ A = \begin{bmatrix} 3 & -1 & -1 \\ 1 & -3 & 3 \end{bmatrix} \] and \[ \vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \] We need to find the solutions for \(\vec{x}\) that satisfy this equation.

To solve \( A \vec{x} = \overrightarrow{0} \), we find the null space of matrix \( A \). The null space consists of all vectors \(\vec{x}\) such that \( A \vec{x} = \overrightarrow{0} \).

Perform row reduction on the matrix \( A \) to find its null space. The row-reduced form of \( A \) reveals that there is one free variable, indicating an infinite number of solutions.

The null space of \( A \) is spanned by the vector \[ \begin{bmatrix} \frac{3}{4} \\ \frac{5}{4} \\ 1 \end{bmatrix} \] This means that the solutions can be expressed in terms of a parameter \( t \) as follows: \[ \vec{x} = t \begin{bmatrix} \frac{3}{4} \\ \frac{5}{4} \\ 1 \end{bmatrix} \] where \( t \) is any real number.

Final Answer