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Find the following determinant. $$\det\begin{bmatrix} 1 & 0 & 2 & -1\\ 0 & 1 & 4 & -2 \\ 2 & -1 & 3 & 1 \\ 2 & -1 & -1 & 2\end{bmatrix}$$


Theoretically I know how to find the determinant of a $m \times n$ matrix, as follow:

  1. Gaussian elimination,
  2. Laplace method,
  3. Gaussian elimination + Laplace method,
  4. Sarrus's rule (this method is valid only for $3 \times 3$ matrix),
  5. $2 \times 2$ matrix (easiest way).

I used the 3rd method (because if I had used only the second one, it would have be a long and tedious process).

here's my attempt to solve it:

  1. I reduced the matrix using Gaussian elimination and the result is as follows:

$$\begin{bmatrix} 1 & 0 & 2 & -1\\ 0 & 1 & 4 & -2 \\ 0 & -1 & -1 & 2 \\ 0 & -1 & -5 & 4\end{bmatrix}$$ This is enough in order to compute the determinant of this matrix, using the first column.

  1. I used Laplace method and here's what I got: $$(-1)^2 \det\begin{bmatrix} 1 & 4 & -2\\ -1 & -1 & 2\\ -1 & -5 & 4\end{bmatrix}$$

My result (using Sarrus's rule to compute the determinant of the minor) is 6, but the exercise has these possible solutions:

(1) -7

(2) 0

(3) 7

(4) 5

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  • $\begingroup$ You must be the only one who can compute the determinant of a non-square matrix. $\endgroup$ – Rodrigo de Azevedo Dec 19 '20 at 12:34
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The $(3,4)$ entry of your matrix after step $1$ should be a $3$.

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  • $\begingroup$ do you mean that step 2 was wrong? $\endgroup$ – Gabriel Burzacchini Dec 19 '20 at 9:17
  • $\begingroup$ @GabrielBurzacchini Error carried forward. $\endgroup$ – Parcly Taxel Dec 19 '20 at 9:17
  • $\begingroup$ Ah, so the gaussian elimination was wrong, Here is what I did: -2*R3 + R3 --> R3, -2*R4 + R4 --> R4. Ah, now I saw the error. Thanks $\endgroup$ – Gabriel Burzacchini Dec 19 '20 at 9:20

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