Answer to Question #203985 in Linear Algebra for Snakho

Question

Inspect the following without finding the determinant:

1.

[1 0 0 0]

[0 1 0 0]

[0 0 1 0]

[0 0 0 - 2]

2. [1 0 0 0]

[0 1 0 0]

[0 0 0 1]

[0 0 1/4 0]

Answer 1

If all elements of a row (or column) of a determinant are multiplied by some scalar number k, the value of the new determinant is k times of the given determinant.

If any two row (or two column) of a determinant are interchanged the value of the determinant is multiplied by $-1.$

In identity matrix $a_{ii}=1,$ and all other elements = 0, hence the determinant is 1.

1.

$$\begin{vmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -2 \\ \end{vmatrix}=-2\begin{vmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{vmatrix}=-2(1)=-2$$

2.

$$\begin{vmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1/4 & 1 \\ \end{vmatrix}=1/4\begin{vmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ \end{vmatrix}$$

$$=-1/4\begin{vmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{vmatrix}=-1/4$$