Question

Find the matrix B if A = ${\displaystyle \left[\begin{array}{cc}4& 1\\ 2& 3\end{array}\right]}$ and A² = A + 2B

Answer 1

A = ${\displaystyle \left[\begin{array}{cc}4& 1\\ 2& 3\end{array}\right]}$

Let  B = ${\displaystyle \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}$

A² = A x A = ${\displaystyle \left[\begin{array}{cc}4& 1\\ 2& 3\end{array}\right]\left[\begin{array}{cc}4& 1\\ 2& 3\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}16+2& 4+3\\ 8+6& 2+9\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}18& 7\\ 14& 11\end{array}\right]}$

A + 2B = ${\displaystyle \left[\begin{array}{cc}4& 1\\ 2& 3\end{array}\right]+2\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}4& 1\\ 2& 3\end{array}\right]+\left[\begin{array}{cc}2a& 2b\\ 2c& 3+2d\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}4+2a& 1+2b\\ 2+2c& 3+2d\end{array}\right]}$
∵ A2 = A + 2B

∴ ${\displaystyle \left[\begin{array}{cc}18& 7\\ 14& 11\end{array}\right]=\left[\begin{array}{cc}4+2a& 1+2b\\ 2+2c& 3+2d\end{array}\right]}$
Comparing the corresponding elements
4 + 2a = 18
⇒ 2a = 18 – 4 = 14
∴ a = 7
1 + 2b = 7
⇒ 2b = 7 – 1 = 6
∴ b = 3
2 + 2c = 14
⇒ 2c = 14 – 2 = 12
∴ c = 6
3 + 2d = 11
⇒  2d = 11 – 3 = 8
∴ d = 4
Hence a = 7, b = 3, c = 6, d = 4
∴ B = ${\displaystyle \left[\begin{array}{cc}7& 3\\ 6& 4\end{array}\right]}$.