algebraic multiplicity of matrix

srinivasanlsn

hi friends plz 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

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
[tex]\left|\begin{array}{cccc}a & b & c & d \\ e & f & g & h \\ i & j & k & l\\ m & n & o & p\end{array}\right|[/tex]
is given by
[tex]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|[/tex]

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.