Is Det(A+B) Equal to Det(A) + Det(B)?

[phyin]

det(A+B) ?? det(A) + det(B)

I'm guessing greater than but I'm not too sure. I need a proof on this so I can be assured of it and then use the statement to prove something else.

any hint (or link to proof) would be much appreciated.

edit:

x*det(A) ?? det(x*A)
what's the relation there?

 

[micromass]

Neither holds, take

$$A=B=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right)$$

for one equality and


$$A=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right),~B=-A$$

for the other.

For the last question, we do have the equality

$$det(xA)=x^n detA$$

where A is the rank of the matrix. From this we can deduce that neither of the inequalities between det(xA) and xdet(A) holds true. Just take x>1 and x<1.

 

[phyin]

thanks. i better find an alternative way to my proof ;s

 

[AlephZero]

Another simple counter-example is
$$A = \begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix} \qquad <br /> B = \begin{pmatrix} 0 & 0 \\ 0 & b \end{pmatrix}$$

det(A) = det(B) = 0, det(A+B) = ab