Linear Algebra: Finding a Polynomial for Matrix A - Help Needed!

[momo20]

Hey,

Could someone help me with my question here please ??

it's a linear algebra que ...

let the (matrix) A= a b
c d

Find a polynomial p(x) = x^2 + qx + r
such that p(A)=0

 

[Dick]

Look up and learn about characteristic polynomials. This question didn't come out of a vacuum. There's a chapter in your book about it.

 

[momo20]

unfortunately not !

nothing in my book says anything about polynomial .. i searched all the day for that !

 

[dynamicsolo]

Hey,

Could someone help me with my question here please ??

it's a linear algebra que ...

let the (matrix) A= a b
c d

Find a polynomial p(x) = x^2 + qx + r
such that p(A)=0

Although it looks strange, you can set the variable in a polynomial equal to a matrix. What would x^2 mean if x = A? The next term would be qA , which has a clear meaning. The last term has to be interpreted in terms of matrices: in numerical algebra, r = r·1 , so in matrix algebra, r = r · I . You can now add the terms of your polynomial, obeying matrix addition. You are looking to express q and r in terms of the entries in A, so that the sum is 0, that is, the zero matrix.

 

[Dick]

Try eigenvalues. The quick story is that A satisfies the polynomial determinant(A-x*I)=0 where I is the identity matrix and x is the variable in your polynomial. Does that sound familiar?

 

[HallsofIvy]

unfortunately not !

nothing in my book says anything about polynomial .. i searched all the day for that !

A linear algebra textbook that doesn't say anything about characteristic polynomials? I find that hard to believe. In fact, from the nature of this problem, I suspect they may be introduced in the very next section: this problem looks like an introduction to them!

But since you have not yet seen characteristic polynomials (and have already spent a day on this), how about the obvious: Go ahead and calculate x², for x equal to this matrix, put it into the equation x²+ qx+ rI (that last term is the constant r multiplying the identity matrix in order to make it a matrix equation) and see what p and q must be in order that it equal the 0 matrix.

You will get four equations for the two variables, p and r, but they are not all independent.