Question

If $A$ is an involutory matrix and $(I+A{)}^n=aI+bA,(I\text{is unit matrix}),a,b\in R.$ Then $(a+b)=$

• A. $2^{n+1}$
• B. $2^n$
• C. $2^{n-1}$
• D. $0$

Answer 1

The correct option is B $2^n$

We know, for involutory matrix of $A:A^{2k}=I$ and $A^{2k+1}=A,k\in N$
Now, $(I+A{)}^n=C_0{I}^n+C_1{I}^{(n-1)}A+C_2{I}^{(n-2)}{A}^2+\cdots +C_n{A}^n=(C_0+C_2+\cdots +C_n)I+(C_1+C_3+\cdots +C_{n-1})A=(2^{n-1})I+(2^{n-1})A$
So, $a=2^{n-1}=b$
$\therefore a+b=2^n$