Question

Construct \(3 \times 2\) matrices \(A\) and \(B\) such that \(Ax = 0\) has only the trivial solution and \(B{\mathop{\rm x}\nolimits} = 0\) has a nontrivial solution.

Short answer

Matrices \(A\) and \(B\) are \(\left[ {\begin{array}{*{20}{c}}1&0\\0&1\\0&0\end{array}} \right],\left[ {\begin{array}{*{20}{c}}1&0\\0&0\\0&0\end{array}} \right]\).

Step-by-step solution

Construct a \(3 \times 2\) matrix \(A\), such that \(Ax = 0\) has only a trivial solution

The columns of matrix \(A\) arelinearly independentif and only if the equation \(Ax = 0\) has only a trivial solution.

Construct any \(3 \times 2\) matrix \(A\) with two non-zero columns, such that neither column is a multiple of the other.

\(A = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\\0&0\end{array}} \right]\)

Construct a \(3 \times 2\) matrix \(B\), such that \(Bx = 0\) has a non-trivial solution

Construct any \(3 \times 2\) matrix \(B\) in which one column is a multiple of the other column.

\(B = \left[ {\begin{array}{*{20}{c}}1&0\\0&0\\0&0\end{array}} \right]\)

Thus, matrices \(A\) and \(B\) are \(\left[ {\begin{array}{*{20}{c}}1&0\\0&1\\0&0\end{array}} \right],\left[ {\begin{array}{*{20}{c}}1&0\\0&0\\0&0\end{array}} \right]\).