Answer to Question #201699 in Linear Algebra for Zoo

Question #201699

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


1
Expert's answer
2021-06-03T16:50:12-0400

(7.1)

1000010000100002\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=1×1×1×(2)Determinant=2Determinant = 1\times 1\times 1\times (-2)\\\boxed{Determinant=-2}



(7.2)

100001000001001/40\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:


10000100001/400001\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=1×1×(1/4)×1Determinant=14Determinant = 1\times 1\times( -1/4)\times 1\\\boxed{Determinant=-\dfrac{1}{4}}


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