I tried the following problem.
Let $A = \begin{bmatrix} \alpha & 0 \\ 0 & \beta \end{bmatrix}$ and $B = \begin{bmatrix} 0 & \gamma \\ \delta & 0 \end{bmatrix}$.
There are 2 statements:
- $AB – BA$ is always an invertible matrix, and
- $AB – BA$ is never an identity matrix.
Now I'm being asked to find out whether these statements are true or false.
I'm not too familiar with $\LaTeX$ and I don't have enough rep to upload images, I had to put my problem as a linked image:
On calculation:
I first got the product of the two matrices as:
$AB = \begin{bmatrix} 0 & \alpha\gamma \\ \beta\delta & 0 \end{bmatrix}$
$BA = \begin{bmatrix} 0 & \beta\gamma \\ \alpha\delta & 0 \end{bmatrix}$
$AB – BA = \begin{bmatrix} 0 & (\alpha-\beta)\gamma \\ (\beta-\alpha)\delta & 0 \end{bmatrix}$
From the above I got that $AB-BA$ cannot be an identity matrix since the main diagonal elements are all zero. So statement 2 is correct, I believe.
And I also felt Statement 1 too was correct, since $|AB-BA|=\gamma\delta(\alpha-\beta)^2$. (Ouchie, it was quite careless of me!)
But the answer tells me a different story.
I deduced that there are two possibilities: either the answer given is incorrect, or I have made an error somewhere and I am not able to identify.
Best Answer
"I deduced that there are two possibilities" : Actually , there are three Possibilities !
Here , answer given is incorrect & OP made mistakes too.
OP is almost right that $AB-BA$ has Determinant $\color{blue}{+}\gamma\delta(\alpha-\beta)^2$ (( the Outer Symbol is $\color{blue}{+}$ , not $\color{red}{-}$ , thanks to new user "kevin martin" who caught that ))
[[ Text Book is wrong that Determinant is $(\alpha-\beta)^2\delta$ ]]
When either $\gamma=0$ or $\delta=0$ or $\alpha=\beta$ , Determinant is $0$ & that Matrix will have no Inverse.
Hence Statement 1 is not true.
OP is right that $AB-BA$ is not $I$.
Text Book is wrong that $AB-BA$ can somehow become $I$
It is never Identity Matrix.
Hence Statement 2 is true.
[[ It might be "Symmetric Matrix" , when $\delta=-\gamma$ ]]
OBSERVATIONS :
Text Book has a lot of typos :
Eg misplaced $)]$ to $])$
Eg mispaced $(\alpha-\beta)^2$ to $(\alpha-\beta^2\rangle$
Text Book has a lot of Assumptions :
Eg Non-Zero Elements
Eg Non-Equal Elements
Text Book is wrong in Basics :
Eg Mixing Identity with Symmetry
Eg Writing $\gamma=-\delta$ AND $\delta=-\gamma$ which is repeating SAME CRITERIA !
SUGGESTION : Use better text book to Improve yourself.
Check out :
https://brilliant.org/wiki/matrices/
https://brilliant.org/courses/linear-algebra/
https://openstax.org/details/books/college-algebra-2e
https://open.umn.edu/opentextbooks/textbooks/213
https://en.wikibooks.org/wiki/Introductory_Linear_Algebra/Matrices
https://en.wikibooks.org/wiki/Introductory_Linear_Algebra/Matrix_inverses_and_determinants