Answer to Question #201699 in Linear Algebra for Zoo

Question

Without calculating the determinant, inspect the following:

(7,1) 1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 −2

(7.2) 1 0 0 0

0 1 0 0

0 0 0 1

0 0 1/4 0

Answer 1

(7.1)

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

Since it is a diagonal matrix.

So, the determinant can be calculated by inspection(by multiplying the diagonal elements)

$Determinant = 1\times 1\times 1\times (-2)\\\boxed{Determinant=-2}$

(7.2)

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

Interchanging third and fourth row , which is a valid transformation with respect to the determinant (it will leave it unchanged), you will get:

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

Since it is a diagonal matrix.

So, the determinant can be calculated by inspection(by multiplying the diagonal elements)

$Determinant = 1\times 1\times( -1/4)\times 1\\\boxed{Determinant=-\dfrac{1}{4}}$