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Algebra Level 5

Let j : R 2 → R 2 j: \mathbb{R^2} \rightarrow \mathbb{R^2} be the linear operator defined by ( x , y ) → ( x + 2 y , − y ) (x,y) \rightarrow (x + 2y, -y) . If A = M a t E ( j ) A = Mat_{E}(j) where E = E = { ( 1 , 0 ) , ( 0 , 1 ) (1,0),(0,1) }. Find − ∣ A ∣ -|A| .


The answer is 1.

Paul Ryan Longhas by Paul Ryan Longhas , 24, Philippines

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1 solution

Matrix A A is the matrix representation of j j with respect of the basis E E so evaluating the linear operator with each value of the basis we get: ( 1 , 0 ) (1,0) and ( 2 , − 1 ) (2,-1) , arranging this to a matrix

A = ∣ 1 2 0 − 1 ∣ A=\begin{vmatrix} 1&2 \\ 0& -1 \end{vmatrix} ⇒ − ∣ A ∣ = − d e t ( A ) = 1 \Rightarrow -\left |A\right |=-det(A)=1

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