# Is the Determinant of a Matrix Additive?

## Problem 186

Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$.

Is it true that
$\det(A+B)=\det(A)+\det(B)?$ If so, then give a proof. If not, then give a counterexample.

Contents

## Solution.

We claim that the statement is false.
As a counterexample, consider the matrices
$A=\begin{bmatrix} 1 & 0\\ 0& 0 \end{bmatrix} \text{ and } B=\begin{bmatrix} 0 & 0\\ 0& 1 \end{bmatrix}.$ Then
$A+B=\begin{bmatrix} 1 & 0\\ 0& 1 \end{bmatrix}$ and we have
$\det(A+B)=\begin{vmatrix} 1 & 0\\ 0& 1 \end{vmatrix}=1.$

On the other hand, the determinants of $A$ and $B$ are
$\det(A)=0 \text{ and } \det(B)=0,$ and hence
$\det(A)+\det(B)=0\neq 1=\det(A+B).$

Therefore, the statement is false and in general we have
$\det(A+B)\neq \det(A)+\det(B).$

## Remark.

When we computed the $2\times 2$ matrices, we used the formula
$\begin{vmatrix} a & b\\ c& d \end{vmatrix}=ad-bc.$

This problem showed that the determinant does not preserve the addition.
However, the determinant is multiplicative.
In general, the following is true:
$\det(AB)=\det(A)\det(B).$

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