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प्रश्न
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
पर्याय
a = 4, b = 1
a = 1, b = 4
a = 0, b = 4
a = 2, b = 4
उत्तर
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 .
\[ \Rightarrow a + 1 = \text{2 and b - 1} = 3\]
\[ \therefore a = \text{1 and b} = 4\]
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