Mathematics - Matrices Question with Solution | TestHub

Given the conditions $AB=A$ and $BA=B$, it follows that $A^2=A$ and $B^2=B$.

Let's expand $(A+B{)}^2$:

$$(A+B{)}^2=A^2+B^2+AB+BA$$

Substitute the given conditions:

$$A^2+B^2+AB+BA=A+B+A+B=2(A+B)$$

Now, let's expand $(A+B{)}^3$:

$$(A+B{)}^3=(A+B{)}^2(A+B)=2(A+B)(A+B)=2(A+B{)}^2$$

Since $(A+B{)}^2=2(A+B)$, we can substitute this back:

$$2(A+B{)}^2=2[2(A+B)]=2^2(A+B)$$

Following this pattern, for $(A+B{)}^n$, the result will be $2^{n-1}(A+B)$.

Therefore, for $(A+B{)}^7$:

$$(A+B{)}^7=2^{7-1}(A+B)=2^6(A+B)=64(A+B)$$