FYI I tried to find related questions using the proper tags and phrases: week 4, practice quiz
can someone rule out that the only justification for making a mistake here is only because the
last sentence didnt say ‘The negative area…’
Hi @Yigal_Ornes
Let’s break down the math step-by-step to see exactly what happened here.
When constructing the transformation matrix A, we place the transformed basis vectors into the columns. The standard matrix is built by mapping T(1,0) to the first column and T(0,1) to the second column:
Now, calculating the determinant of this matrix gives:
\\det(A) = (3 × 5) - (2 ×1) = 15 - 2 = 13
Notice that the determinant is actually positive 13.
Why did you get a negative value?
It is highly likely you swapped the column order when setting up the matrix, putting T(0,1) first:
Calculating the determinant of this swapped matrix gives
(2 ×1) - (3× 5) = 2 - 15 = -13.
Swapping two columns always flips the sign of the determinant because it reverses the vector orientation.
Regardless of the column order, geometric area is strictly an absolute value
(|-13| = 13).
The question asks for the geometric area spanned by the transformation, which is a physical measurement of space and can never be negative.
So the text is fully accurate—no need for a “negative area” disclaimer!
Thanks for replying
okay the word geometric doesnt appear
but i see, if they wanted a determinant they would use that word ‘determinant’
I refer you to week 4: ‘determinant as an area’
to get the exact words the lecture uses…
this issue was not about material clarity but about phrasing that for me seems a bit ambiguous
the order of base vectors that is known from lecture to affect the sign (for non singular) is presented in the question in 2 different ways, now if you say area as in geometric area…its seems also ambiguous in regard to lectures…
I see exactly what you mean now.
You are totally right about the video , the lecture does teach that changing the order of the basis vectors gives a “signed” or negative area depending on the orientation.
The issue here is a slight disconnect between the lecture’s phrasing and the quiz logic. While the video uses the concept of negative area to explain orientation, standard quiz questions on this topic usually expect the absolute magnitude whenever they ask for the final “Area” value.
Your point is completely valid. The phrasing is definitely ambiguous given how the concept is introduced in the video, and it is very fair to feel that the question should be more precise.
The instructor uses a lot of non standard terminology
As I replied to Omar, the lectures clearly present the material and one can understand what Omar said about the direction or shape from the lecture…
I failed because i assumed area as in determinant is what they asked and i deduced it because following the text, that there order of basis vectors is important…
now i am thinking that because ambiguity exists maybe, this is the sign to actually give the geometric area 
last emphasis from me…i was aiming to understand on how to read the question properly for future assignments (and courses), i felt i am missing something and couldnt resolve it
completely understand, and you didn’t miss anything since you followed the lecture perfectly.
Your logic was solid, and knowing this will help you read the next questions easily!