Question

Suppose $A$ and $B$ are two non singular matrices such that $B\ne I,A^6=I$ and $A{B}^2=BA$. Find the least value of $k$ for $B^k=I$.

Answer 1

Step 1: Given:

$|A|=|B|\ne 0$

$A^6=I$

$A{B}^2=BA$

$B^k=I$

Step 2: To find:

Value of $k$ for which $B^k=I$

Step 3: Formula used:

$A{A}^{-1}=A^{-1}A=I$

Step 4: Solution:

Consider the given expression

$A{B}^2=BA$ ... (1)

Step 5: Aim to make RHS as independent of $A$

Pre-multiply (1) from both sides by $A^5$, we get

$A^6{B}^2=A^{5}BA$

Post-multiply above equation from both sides by $A^5$, we get

$A^6{B}^2{A}^5=A^{5}B{A}^6$

Since, it is given that $A^6=I$

Expression (3) reduces to

$B^2{A}^5=A^{5}B$

Pre-multiply this on both sides by $A$, we get

$A{B}^2{A}^5=A^{6}B$

$\Rightarrow A{B}^2{A}^5=B$ $(A^6=I)$ .....(2)

Step 8:

Now substituting the value of $B$ from (2) in LHS of (2) itself, we get

$A(A{B}^2{A}^5{)}^2{A}^5=B$

$\Rightarrow AA{B}^2{A}^{5}A{B}^2{A}^5{A}^5=B$

$\Rightarrow A^2{B}^2{A}^6{B}^2{A}^{10}=B$

Since, we know $A^6=I$

$\Rightarrow A^2{B}^4{A}^6{A}^4=B$

$\Rightarrow A^2{B}^4{A}^4=B$ ... (3)

Step 9:

Now substituting the value of $B$ from (2) in LHS of (3), we get

$A^2(A{B}^2{A}^5{)}^4{A}^4=B$

$\Rightarrow A^{2}A{B}^2{A}^{5}A{B}^2{A}^{5}A{B}^2{A}^{5}A{B}^2{A}^5{A}^4=B$

$\Rightarrow A^3{B}^2{A}^6{B}^2{A}^6{B}^2{A}^6{B}^2{A}^5{A}^4=B$

Since, we know $A^6=I$

$\Rightarrow A^3{B}^8{A}^3=B$ ...(4)

Step 10:

Looking at the pattern of (2), (3) and (4), we can say that

$A^p{B}^q{A}^r=B$

$p$ is increasing by $1$,

$q$ is forming a GP as $2,4,8,...$, and

$r$ is decreasing by $1$

$\therefore$ In the same manner, we find that when $p=6$, we get $q=64$ and $r=0$

So, the expression becomes

$A^6{B}^{64}{A}^0=B$

Since, we know $A^6=I$

$\Rightarrow B^{64}=B$ ...(5)

Step 11:

Now, we have $B^{64}=B$

Post-Multiplying this by $B^{-1}$ both sides, we get

$B^{63}B{B}^{-1}=B{B}^{-1}$ ... (6)

Since, $B{B}^{-1}=I$, (6) reduces to

$B^{63}=I$ .... (7)

Step 12:

Comparing (7) with the given condition where $B^k=I$, we find that

$B^{63}=B^k$

$\Rightarrow k=63$

Step 13: Result:

$k=63$