Question

Suppose \(A\) is a \(5 \times 3\) matrix and there exists a \(3 \times 5\) matrix \(C\) such that \(CA = {I_3}\). Suppose further that for some given b in \({\mathbb{R}^5}\), the equation \(A{\mathop{\rm x}\nolimits} = b\) has at least one solution. Show that this solution is unique.

Short answer

It is proved that the solution of the equation \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm b}\nolimits} \) is unique.

Step-by-step solution

Show that the solution is unique

According to the hypothesis, A is a \(5 \times 3\) matrix, \(C\) is a \(3 \times 5\) matrix, and \(AC = {I_3}\). Let \(x\) satisfy \(A{\mathop{\rm x}\nolimits} = b\). Then, \[CA{\mathop{\rm x}\nolimits} = C{\mathop{\rm b}\nolimits} \].

Also, x must be \(C{\mathop{\rm b}\nolimits} \) because \(CA = I\). Hence, \(C{\mathop{\rm b}\nolimits} \) is the only solution of \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm b}\nolimits} \).

Thus, it is proved that the solution of the equation \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm b}\nolimits} \) is unique.