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How do you find ${{B}^{-1}}$?; We know that ${{B}^{2}}=B+2{{I}_{3}}$.

Last updated date: 15th Jul 2024
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Hint: In this problem, we have to find the inverse of B with the given condition ${{B}^{2}}=B+2{{I}_{3}}$. We can first multiply ${{B}^{-1}}$ on both the left-hand side and the right-hand side of the given equation. We can then substitute equivalent values for each term, we will get a term ${{B}^{-1}}$, we can take the remaining terms to the other side to find the value of ${{B}^{-1}}$.

We know that the given equation is,
${{B}^{2}}=B+2{{I}_{3}}$….. (1)
We have to find ${{B}^{-1}}$.
We can multiply ${{B}^{-1}}$ on on both the left-hand side and the right-hand side of the equation (1), we get
$\Rightarrow {{B}^{2}}{{B}^{-1}}=B{{B}^{-1}}+2{{I}_{3}}{{B}^{-1}}$
We can now write the left-hand side as,
$\Rightarrow \dfrac{{{B}^{2}}}{B}=B{{B}^{-1}}+2{{I}_{3}}{{B}^{-1}}$
We can now cancel the terms in the left-hand side, we get
$\Rightarrow B=B{{B}^{-1}}+2{{I}_{3}}{{B}^{-1}}$…… (2)
We can now simplify the right-hand side.
We can write $B{{B}^{-1}}={{I}_{3}}$ and ${{I}_{3}}{{B}^{-1}}={{B}^{-1}}$.
We can now substitute the above values in the equation (2), we get
$\Rightarrow B={{I}_{3}}+2{{B}^{-1}}$
We can now subtract ${{I}_{3}}$ on both the left-hand side and the right-hand side.
$\Rightarrow B-{{I}_{3}}=2{{B}^{-1}}$
Now we can divide the number 2 on both right-hand side and the left-hand side, we get
$\Rightarrow \dfrac{1}{2}\left( B-{{I}_{3}} \right)={{B}^{-1}}$

Therefore, the value of ${{B}^{-1}}=\dfrac{1}{2}\left( B-{{I}_{3}} \right)$.

Note: Students make mistakes while substituting the equivalent values to simplify the steps, like substituting $B{{B}^{-1}}={{I}_{3}}$ and ${{I}_{3}}{{B}^{-1}}={{B}^{-1}}$. We had divided the number 2 in the above step on the both right-hand side and left-hand side, as we have to find the value of ${{B}^{-1}}$.