Find unkown value for matrices

I am new at matrices and algebra. Kindly help me out to solve the problem from the first chapter of my textbook.

For a lower triangular matrix L and an upper triangular matrix U of order three with unit diagonal elements determine P such that

(a) LP = PL = I

(b) UP = PU = I

Re: Find unkown value for matrices

To transform a matrix $\displaystyle A$ to the identity, $\displaystyle I$, you multiply it by the inverse $\displaystyle A^{-1}$.

$\displaystyle A A^{-1} = A^{-1} A = I $

So the question is really, "Find the inverse of L and of U."

Re: Find unkown value for matrices

Quote:

Originally Posted by

**learncsmath** I am new at matrices and algebra. Kindly help me out to solve the problem from the first chapter of my textbook.

For a lower triangular matrix L and an upper triangular matrix U of order three with unit diagonal elements determine P such that

(a) LP = PL = I

(b) UP = PU = I

Are you **given** specific matrices L and U? This seems a very general problem which is going to have a very complicated solution if the off diagonal elements of L and U are "generic". Further, as Fernando Revilla says, you are really asked for a matrix that is the inverse of both L and U. For general L and U, that will not exist.

Re: Find unkown value for matrices

Actually, this problem is *not* that hard, provided it is two separate problems. The second one, for example, is to find U^{-1}, where:

$\displaystyle U = \begin{bmatrix}1&a&b\\0&1&c\\0&0&1 \end{bmatrix}$.

you should know (hopefully) that the product of two upper-triangular matrices with diagonal entries of 1, is another such matrix. since I (the identity matrix) is also upper triangular, with diagonal entries all 1, it seems reasonable to conjecture that U^{-1} is also upper-triangular with 1's on the diagonal. so we want to solve THIS equation:

$\displaystyle \begin{bmatrix}1&a&b\\0&1&c\\0&0&1 \end{bmatrix} \begin{bmatrix}1&x&y\\0&1&z\\0&0&1 \end{bmatrix} = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1 \end{bmatrix}$

for x,y and z, in terms of a,b and c. you can do this.

Re: Find unkown value for matrices

Quote:

Originally Posted by

**Deveno** Actually, this problem is *not* that hard, provided it is two separate problems. The second one, for example, is to find U^{-1}, where:

$\displaystyle U = \begin{bmatrix}1&a&b\\0&1&c\\0&0&1 \end{bmatrix}$.

you should know (hopefully) that the product of two upper-triangular matrices with diagonal entries of 1, is another such matrix. since I (the identity matrix) is also upper triangular, with diagonal entries all 1, it seems reasonable to conjecture that U^{-1} is also upper-triangular with 1's on the diagonal. so we want to solve THIS equation:

$\displaystyle \begin{bmatrix}1&a&b\\0&1&c\\0&0&1 \end{bmatrix} \begin{bmatrix}1&x&y\\0&1&z\\0&0&1 \end{bmatrix} = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1 \end{bmatrix}$

for x,y and z, in terms of a,b and c. you can do this.

Thanks for the reply. I have solved the problem. As a beginner I was not aware that the product of two upper triangular matrices is also an upper triangular matrix. Keeping that concept in mind I solved for P and got the correct answer. Thanks again.

Re: Find unkown value for matrices