MHB Checking Invertibility of A^3 Matrix - Tips & Solutions

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Hi all,

I have the matrix A shown in the attached photo.

View attachment 3667

I need to check for which values of k, the matrix A^3 is invertible.

I tried calculating A^3, by multiplying A by itself twice. I got a nasty matrix. It makes no sense to me that now I am suppose to find it's rank, by using operations on rows. Am I missing something ? How would you solve this in the easiest way ?

Thanks !
 

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Hi Yankel,

Note $A^3$ is invertible if and only if $A$ is invertible. How would you determine the values of $k$ that make $A$ invertible?
 
Oh, I see, this was the missing part.

To find values for which A is invertible is easier. I find the rank of A, right ?
 
No, finding the rank of $A$ is unnecessary. If you can use determinants, find the values of $k$ for which $\text{det}(A) \neq 0$. If you don't know determinants, then consider row reducing the augmented matrix $[A|I]$ to Gaussian form.
 
The world of 2\times 2 complex matrices is very colorful. They form a Banach-algebra, they act on spinors, they contain the quaternions, SU(2), su(2), SL(2,\mathbb C), sl(2,\mathbb C). Furthermore, with the determinant as Euclidean or pseudo-Euclidean norm, isu(2) is a 3-dimensional Euclidean space, \mathbb RI\oplus isu(2) is a Minkowski space with signature (1,3), i\mathbb RI\oplus su(2) is a Minkowski space with signature (3,1), SU(2) is the double cover of SO(3), sl(2,\mathbb C) is the...
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