Thread: a question on 4x4 matrix

A and B are real 4x4 matrices, if det(B)=1 and B=2A^3(A)T (T means transpose), what is the det(A)? Thank you very much in advance for your time and effort.

From how I interpreted your problem:

$\displaystyle det(B) = 1$
$\displaystyle B = 2A^3A^T $

If that's the case, remember determinants can be broken up over multiplication.

So if I re-write B:

$\displaystyle B = 2A^3A^T = 2 A A A A^T $

Now that you have that, be careful when you multiply by a scalar (in this case 2). Think about how that changes the matrix A in terms of row operations.

Hi eXist,

Thanks very much for your reply. Could you please explain further as I am a newbie to matrix. from det(B)=1, I can see B is an identity matrix and det(2AAA(A)T)=1. so how can I solve det(A)? Do I need to find the matrix A? Thank you and please excuse me for my poor knowledge on this.

cheers,
pingping

Originally Posted by shenpingping

Hi eXist,

Thanks very much for your reply. Could you please explain further as I am a newbie to matrix. from det(B)=1, I can see B is an identity matrix and det(2AAA(A)T)=1. so how can I solve det(A)? Do I need to find the matrix A? Thank you and please excuse me for my poor knowledge on this.

cheers,
pingping

From what you have been told you should see that:

$\displaystyle 2^n [\det(A)]^3\det(A^T)=1$

also you should know that $\displaystyle \det(A^T)=\det(A)$ so:

$\displaystyle 2^n [\det(A)]^4=1$

where $\displaystyle n$ is the size of the matrix in this case $\displaystyle n=4$

(when you multiply a matrix by a scalar every element is multiplied by the scalar, and for an $\displaystyle n \times n$ matrix the determinant can be written as a sum of $\displaystyle \pm$ the product of $\displaystyle n$ terms at a time from the matrix, this is where the $\displaystyle 2^n$ term above comes from)

CB

Originally Posted by CaptainBlack

$\displaystyle 2^n [\det(A)]^4=1$

where $\displaystyle n$ is the size of the matrix in this case $\displaystyle n=4$

Yep, CB beat me to it. Just wanted you to see that multiplying a matrix by a scalar, changes the determinant of the matrix by that scalar to the nth power. n being the size of the matrix.

Thanks CB.
-Chad

Hi CaptainBlack,

Thanks you very much for your explanation. greatly appreciate.

cheers,
pingping