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Question

# True or False: Given any real number r and any$3 \times 3$matrix A whose entries are all nonzero, it is always possible to change at most one entry of A to get a matrix B with det(B) = r.

Solution

Verified

The statement is false.

Consider the matrix

$A=\left[\begin{array}{ccc} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 1 & 1\end{array}\right]$

.

We have that $\det(A)=0$ since $A$ has equal rows.

Observe that replacing the entry $a_{ij}$ of $A$ by any real number $x$ we obtain the matrix $A_{ij}(x)$ which still has two equal rows and therefore $\det(A_{ij}(x))=0$.

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