Algebraic multiplicity of matrix

[srinivasanlsn]

hi friends please help in finding out the ans for this . A 3x3 matrix was given , am asked to find algebraic multiplicity of it ! how to find algebraic multiplicity of 3x3 matrix ??

 

[micromass]

Did you mean the algebraic multiplicity of an eigenvalue??

Well, how did you define it??

 

[srinivasanlsn]

the algebraic multiplicity of the matrix a=[ 0 1 0 ]
[ 0 0 1 ]
[ 1 -3 3 ]
a.1
b.2
c.3
d.4

i don get the question first, somebody help me...

 

[srinivasanlsn]

my next question is how to find determinant of 4x4 matrix ??

 

[srinivasanlsn]

the algebraic multiplicity of the matrix
[ 0 1 0 ]
[ 0 0 1 ]
[ 1 -3 3 ]

options :
a.1
b.2
c.3
d.4

i don get the question first, somebody help me...

 

[micromass]

Again, what is your definition of algebraic multiplicity?

 

[uponit12]

First rearrange and the answer is obvious.

[1 -3 3]
[0 1 0]
[0 0 1]

From here the eigenvalues are obviously [1,1,1]. From here the question says what is the algebraic multiplicity. The question was obviously used for simplicity, so you know the multiplicity for the eigenvalue 1 is 3 since it appears in the diagonal 3 times.

Since it is the only one the answer can only be C.

 

[AlephZero]

We know what the algebraic multiplicity of an eigenvalue is.

The OP's question was about the algebraic multiplicity of a matrix, which is not a term that I have ever seen before (and neither has Google).

Maybe something got "lost in tanslation" here...

 

[HallsofIvy]

my next question is how to find determinant of 4x4 matrix ??

To find the determinant of a 4x4 matrix, you could use the basic definition- but that's very difficult. Most people use "expansion by minors".
The determinant of
$$\left|\begin{array}{cccc}a & b & c & d \\ e & f & g & h \\ i & j & k & l\\ m & n & o & p\end{array}\right|$$
is given by
$$a\left|\begin{array}{ccc}f & g & h \\ j & k & l \\ n & o & p\end{array}\right|- b\left|\begin{array}{ccc}e & g & h \\ i & k & l \\ m & o & p\end{array}\right|+ c\left|\begin{array}{ccc}e & f & h \\ i & j & l\\ m & n & p\end{array}\right|- d\left|\begin{array}{ccc}e & f & g \\ i & j & k \\ n & o & p \end{array}\right|$$

 

[uponit12]

AlephZero & srinivasanlsn,

Remember this post shows that the person asking is answering a multiple choice question, as everyone is aware, which has a common instructor imposed complication written in, which is: as you mentioned a term which has no direct definition but instead must be understood only be really understanding the term multiplicity as it is used in Linear algebra.

The multiplicity of an eigenvalue λ of a linear transformation T as the number of independent associated eigenvectors. That is, as the dimension of the kernel of T-λ1V. etc, etc...

If the student knows this or something similar then the seemingly confusing terminology is made clear by using the terms interchangeably due to a nesting effect of definitions. Much like the use of dimension of the kernel above could be stated in a more confusing manner like using it to ask the question: what is the dimension of the kernel for A=...

Not preaching, as nearly all people answering questions knows this already, just telling the person asking to look through the questions by understanding the definitions better.