Question

$\mathit{d}\mathit{e}\mathit{t}\left(A+B\right)\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{+}\mathit{d}\mathit{e}\mathit{t}\mathit{B}$for all$\mathbf{5}\mathbf{\times }\mathbf{5}$ matricesA andB.

Short answer

Therefore, the given condition is not satisfied. So, the given statement is false.

Step-by-step solution

Definition. 

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

To check whether the given condition is true or false. 

Let,

$A=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$

$B=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]$

It applies

$detA=detB=0$,

But

$$\begin{gathered} det(A+B)=det{I}_5 \\ det(A+B)=1 \end{gathered}$$

Therefore, the given condition is not satisfied. So, the given statement is false.