First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightMatrices And Determinantschevron_rightExercise 4.1

Solution

Given:

AB=B,\qquad BA=A

We need to show:

A^2+B^2=A+B

Multiply $BA=A$ on the left by $A$ :

A(BA)=AA\Rightarrow (AB)A=A^2

But $AB=B$ , so:

(AB)A=BA Rightarrow A^2=BA=A

Similarly, multiply $AB=B$ on the left by $B$ :

B(AB)=BB\Rightarrow (BA)B=B^2

But $BA=A$ , so:

(BA)B=AB\Rightarrow B^2=AB=B

Therefore:

A^2+B^2=A+B

\boxed{A^2+B^2=A+B}