Question

If \[A = \begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}, B = \begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix}\]and (A + B)² = A² + B²,   values of a and b are

Options

• a = 4, b = 1

• a = 1, b = 4 

• a = 0, b = 4

•  a = 2, b = 4

Answer 1

a = 1, b = 4 

\[Here, \]

\[ \left( A + B \right)^2 = A^2 + B^2 \]

\[ \Rightarrow A^2 + AB + BA + B^2 = A^2 + B^2 \]

\[ \Rightarrow AB + BA = O\]

\[ \Rightarrow AB = - BA\]

\[ \Rightarrow \begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}\begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix} = - \begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix}\begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}\]

\[ \Rightarrow \begin{bmatrix}a - b & 2 \\ 2a - b & 3\end{bmatrix} = - \begin{bmatrix}a + 2 & - a - 1 \\ b - 2 & - b + 1\end{bmatrix}\]

\[ \Rightarrow \begin{bmatrix}a - b & 2 \\ 2a - b & 3\end{bmatrix} = \begin{bmatrix}- a - 2 & a + 1 \\ b + 2 & b - 1\end{bmatrix}\]

The corresponding elements of two equal matrices are equal . 

\[ \Rightarrow a + 1 =  \text{2 and  b - 1} = 3\]

\[ \therefore a = \text{1 and b} = 4\]