# I've a small problem with this matrix (to find the determinant)

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

• You must be the only one who can compute the determinant of a non-square matrix. – Rodrigo de Azevedo Dec 19 '20 at 12:34

The $$(3,4)$$ entry of your matrix after step $$1$$ should be a $$3$$.