Question

Let v₁, . . . , v₉ be the column vectors of a non-zero 9 × 9 real matrix A. Let a₁, . . . , a₉ ∈ \(\R\), not all zero, be such that \(\sum^9_{i=1}a_iv_i=0\). Then the system \(Ax=\sum^9_{i=1}v_i\) has

• A. no solution
• B. a unique solution
• C. more than one but only finitely many solutions
• D. infinitely many solutions

Answer 1

The Correct Option is D

The problem involves understanding the solution space for a given system of equations in terms of linear algebra concepts.

Given a 9 × 9 real matrix \(A\) with column vectors \(v_1, v_2, \ldots, v_9\) such that there exist non-zero scalar coefficients \(a_1, a_2, \ldots, a_9\) satisfying:

\(\sum^9_{i=1} a_i v_i = 0\)

This indicates that the vectors \(v_1, v_2, \ldots, v_9\) are linearly dependent, as there is a non-trivial linear combination of them that equals the zero vector. For a square matrix, linear dependence of columns means the matrix is not of full rank and hence not invertible.

The system \(Ax = \sum^9_{i=1} v_i\) can be expressed in terms of the matrix equation:

\(Ax = \mathbf{v}\), where \(\mathbf{v} = v_1 + v_2 + \cdots + v_9\)

Since the matrix \(A\) is not full rank (due to the linear dependence as shown by the non-zero solution to \(\sum^9_{i=1} a_i v_i = 0\)), the system \(Ax = \mathbf{v}\) does not have a unique solution. Instead, the solution is not unique because the rank of \(A\) is less than 9, indicating there are free variables.

If the system is consistent (which it is, as \(\mathbf{v}\) lies in the column space of \(A\)), it will have infinitely many solutions. Free variables corresponding to the null space provide the degrees of freedom leading to an infinite set of solutions.

Thus, the correct answer is that this system has infinitely many solutions.

This follows because:

• The columns of \(A\) are linearly dependent, suggesting the matrix is not full rank.
• A consistent system \(Ax = \mathbf{b}\) with a non-invertible matrix \(A\) has infinitely many solutions if \(\mathbf{b}\) is in the column space of \(A\).
• The solution space dimension is increased by the rank deficiency, yielding infinite solutions when the system is consistent.