Given $\det(A+B) = 0$ or $\det(AB) = 0$, what can be said for $\det(A)$ and $\det(B)$?

Question

Given $\det(A+B) = 0$, what can be said for $\det(A)$ and $\det(B)$?

I was thinking that one is maybe the inverse of the other? Im honestly so confused and new at this :c

Given $\det(AB) = 0$, what can you say for $\det(A)$ and $\det(B)$?

for the second part I wrote that if $A$ and $B$ are square matrices then one of them has $\det$ $0$, if they are not the $\det$ is undefined. is that right? thanks in advance

Answer 1

You can say nothing for the first question.

• Both can be invertibile if say $A=I$ and $B=-I$.
• No one is invertibile if say $A=0=B$.
• One is and second is not invertibile if say $A = I$ and $B= \pmatrix{-1&0\\0&0}$.

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For the second, since we have $\det(AB)= \det(A)\det(B)$ then clearly one is not invertibile.

Answer 2

The answer to the second is correct. The determinant is only defined for square matrices, and $\det(AB) = \det(A)\det(B)$, so if $\det(AB)=0,$ one of $\det(A)$ or $\det(B)$ is $0$.

As for sums, in general, you can't say anything about $\det(A)$ or $\det(B)$ given knowledge of $\det(A+B)$. Take for example $$ A = \left[\begin{array}{cccc} 2 & 0&0&0 \\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{array}\right] \qquad \textrm{and} \qquad B=\left[\begin{array}{cccc} 0&0&0&2\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0 \end{array}\right] $$ Both have a determinant of $2$, but $\det(A+B)=0$. This will in fact be the case no matter what the top left entry of $A$ or the top right entry of $B$ is.

Answer 3

1) If it is given det(A+B)=0 it is not necessary that det(A)=0 or det(B)=0 independently.

2) If det(AB)=0 Either det(A)=0 or det(B)=0 as rules of Matrices clearly state det(AB)=det(A)*det(B)