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Suppose \(CA = {I_n}\)(the \(n \times n\) identity matrix). Show that the equation \(Ax = 0\) has only the trivial solution. Explain why Acannot have more columns than rows.

Short Answer

Expert verified

Amust have at least as many rows as columns because each pivot is in a different row.

Step by step solution

01

The equation \(Ax = 0\) has the trivial solution

When \(x\) satisfies the equation \(Ax = 0\), then \(CA{\mathop{\rm x}\nolimits} = C \times 0 = 0\). So, \({I_n}{\mathop{\rm x}\nolimits} = 0\) and \({\mathop{\rm x}\nolimits} = 0\). This demonstrates that there are no free variables in the equation \(Ax = 0\).

02

Explanation for A cannot have more columns than rows

Therefore, every variable is a basic variable since the equation \(Ax = 0\) has no free variables. Each column of Ais a pivot column. Thus, Amust have at least as many rows as columns because each pivot is in a different row.

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Most popular questions from this chapter

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Solve the equation \(AB = BC\) for A, assuming that A, B, and C are square and Bis invertible.

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