Question

There exist invertible$\mathbf{2}\mathbf{\times }\mathbf{2}$matrices A andBsuch that$\mathit{d}\mathit{e}\mathit{t}\left(A+B\right)\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{+}\mathit{d}\mathit{e}\mathit{t}\mathit{B}$ .

Short answer

Therefore,

$det(\mathrm{A}+\mathrm{B})=detA+detB$.

Yes the statement is true.

Step-by-step solution

Matrix Definition

Matrix is aset of numbers arranged in rows and columns so as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “ m by n ” matrix, written “$m\times n$ .”

To check whether the given condition is true or false

Let,

For example,

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

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

Indeed,

$det\left(A+B\right)=\left[\begin{array}{cc}1& 0\\ 1& 2\end{array}\right]$

$\det (A+B)=2$

$det(A+B)=detA+detB$

Yes the statement is true.