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So I'm asked to proof for any matrix that complies with one of the following rules, the determinant of that matrix is $0$.

  • Two columns of a matrix A are the same
  • A row of a matrix A is a scalar multiple of another row of matrix A
  • The sum of two rows of a matrix A equals a third row of the matrix A

I've been able to prove these rules for both $2\times2$ and a $3 \times 3$ matrices (only $3 \times 3$ in case of the last one), but I'm now questioning how I can extend this to be proven for any $n \times n$ matrix. Is the best option I have here induction or is there an other way to elegantly prove this.

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  • $\begingroup$ What definition do you have for the determinant of a matrix? $\endgroup$ Jun 3, 2020 at 14:51
  • $\begingroup$ Have you already proven that if the determinant of a matrix is non-zero, then that matrix has an inverse? $\endgroup$ Jun 3, 2020 at 14:52
  • $\begingroup$ @Gribouillis, the definition of a matrix here is: $$ \begin{pmatrix} a_{11} & a_{12} & ... a_{1n} \\ a_{21} & a_{22} & ... a_{2n} \\ ... & ... & ... ... \\ a_{m1} & a_{m2} & ... a_{mn} \\\end{pmatrix} $$ $\endgroup$ Jun 3, 2020 at 14:57
  • $\begingroup$ @Omnomnomnom yes I have proven that already $\endgroup$ Jun 3, 2020 at 14:59

1 Answer 1

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I will assume that you have already proven that if the determinant of a matrix is non-zero, then that matrix has an inverse. Recall that if $M$ is invertible, then $Mx = 0$ implies that $x = 0$. With that, note the following:

  • Two columns of $A$ are the same if and only if there is a non-zero column-vector $x$ for which $Ax = 0$, and $x$ the entries of $x$ are all zero except for one entry equal to $1$ and another equal to $-1$.

  • One row is a scalar multiple of the other if and only if there is a non-zero column-vector $x$ for which $A^Tx = 0$ (where $A^T$ denotes the transpose of $A$), and $x$ has only $2$ non-zero entries.

  • The sum of two rows is equal to a third if and only if there is a non-zero column vector $x$ for which $A^Tx = 0$, and $x$ has exactly $3$ non-zero entries, with one of those non-zero entries equal to $1$.


We could prove this as a consequence of the effects of elementary row-operations and column-operations on the determinant of the matrix. In particular, note that

  • Switching two identical columns must change the sign of the determinant, yet the determinant must remain the same,
  • Subtracting a scalar multiple of one row from the other (to produce a zero-row) does not alter the determinant,
  • Subtracting one row from the other (to produce a repeated row) does not alter the determinant.
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