Question #203985

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]


1
Expert's answer
2021-06-07T14:37:50-0400

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.-1.


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


1.


1000010000100002=21000010000100001=2(1)=2\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.


100001000001001/41=1/41000010000010011\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}


=1/41000010000100001=1/4=-1/4\begin{vmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{vmatrix}=-1/4


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