(D) : According to question, A and B are invertible matrices it means - \((\mathrm{AB})^{-1}=\mathrm{B}^{-1} \mathrm{~A}^{-1} \tag{i}\) We know that, \(\mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}\) \(\operatorname{adj} \mathrm{A}=|\mathrm{A}| \mathrm{A}^{-1} \tag{ii}\) Also, \(\quad \operatorname{det}\left(\mathrm{A}^{-1}\right)=[\operatorname{det}(\mathrm{A})]^{-1}\) \(\operatorname{det}(A)^{-1}=\frac{1}{[\operatorname{det}(A)]} \tag{iii}\) \(\operatorname{det}(\mathrm{A}) \cdot \operatorname{det}\left(\mathrm{A}^{-1}\right)=1\) These are true. Again, \((\mathrm{A}+\mathrm{B})^{-1}=\frac{\operatorname{adj}(\mathrm{A}+\mathrm{B})}{|\mathrm{A}+\mathrm{B}|}\) \((\mathrm{A}+\mathrm{B})^{-1} \neq \mathrm{B}^{-1}+\mathrm{A}^{-1} \tag{iv}\)
Karnataka CET-2021
Matrix and Determinant
78780
If \(A=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]\), then \(A^{4}\) is equal to
78782
If \(\left[\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}2 \\ 4\end{array}\right]\), then the values of \(x\) and \(y\) respectively are
1 \(–3, –1\)
2 \(1, 3\)
3 \(3,1\)
4 \(-1,3\)
Explanation:
(D) : \({\left[\begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array}\right]_{2 \times 2}\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \end{array}\right]_{2 \times 1}=\left[\begin{array}{l} 2 \\ 4 \end{array}\right]}\) \({\left[\begin{array}{ccc} \mathrm{x} & +\mathrm{y} \\ -\mathrm{x} & +\mathrm{y} \end{array}\right]=\left[\begin{array}{l} 2 \\ 4 \end{array}\right]}\) On comparing corresponding elements on both the side we get - \(x+y=2 \tag{i}\) \(-x+y=4\) On adding equation (i) and (ii), we get - \(2 y=6 \Rightarrow y=3 \tag{ii}\) Putting in equation, we get - \(x+3=2\) \(x=-1\) The value of \(x\) and \(y\) respectively are \((-1,3)\)
(D) : According to question, A and B are invertible matrices it means - \((\mathrm{AB})^{-1}=\mathrm{B}^{-1} \mathrm{~A}^{-1} \tag{i}\) We know that, \(\mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}\) \(\operatorname{adj} \mathrm{A}=|\mathrm{A}| \mathrm{A}^{-1} \tag{ii}\) Also, \(\quad \operatorname{det}\left(\mathrm{A}^{-1}\right)=[\operatorname{det}(\mathrm{A})]^{-1}\) \(\operatorname{det}(A)^{-1}=\frac{1}{[\operatorname{det}(A)]} \tag{iii}\) \(\operatorname{det}(\mathrm{A}) \cdot \operatorname{det}\left(\mathrm{A}^{-1}\right)=1\) These are true. Again, \((\mathrm{A}+\mathrm{B})^{-1}=\frac{\operatorname{adj}(\mathrm{A}+\mathrm{B})}{|\mathrm{A}+\mathrm{B}|}\) \((\mathrm{A}+\mathrm{B})^{-1} \neq \mathrm{B}^{-1}+\mathrm{A}^{-1} \tag{iv}\)
Karnataka CET-2021
Matrix and Determinant
78780
If \(A=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]\), then \(A^{4}\) is equal to
78782
If \(\left[\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}2 \\ 4\end{array}\right]\), then the values of \(x\) and \(y\) respectively are
1 \(–3, –1\)
2 \(1, 3\)
3 \(3,1\)
4 \(-1,3\)
Explanation:
(D) : \({\left[\begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array}\right]_{2 \times 2}\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \end{array}\right]_{2 \times 1}=\left[\begin{array}{l} 2 \\ 4 \end{array}\right]}\) \({\left[\begin{array}{ccc} \mathrm{x} & +\mathrm{y} \\ -\mathrm{x} & +\mathrm{y} \end{array}\right]=\left[\begin{array}{l} 2 \\ 4 \end{array}\right]}\) On comparing corresponding elements on both the side we get - \(x+y=2 \tag{i}\) \(-x+y=4\) On adding equation (i) and (ii), we get - \(2 y=6 \Rightarrow y=3 \tag{ii}\) Putting in equation, we get - \(x+3=2\) \(x=-1\) The value of \(x\) and \(y\) respectively are \((-1,3)\)
(D) : According to question, A and B are invertible matrices it means - \((\mathrm{AB})^{-1}=\mathrm{B}^{-1} \mathrm{~A}^{-1} \tag{i}\) We know that, \(\mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}\) \(\operatorname{adj} \mathrm{A}=|\mathrm{A}| \mathrm{A}^{-1} \tag{ii}\) Also, \(\quad \operatorname{det}\left(\mathrm{A}^{-1}\right)=[\operatorname{det}(\mathrm{A})]^{-1}\) \(\operatorname{det}(A)^{-1}=\frac{1}{[\operatorname{det}(A)]} \tag{iii}\) \(\operatorname{det}(\mathrm{A}) \cdot \operatorname{det}\left(\mathrm{A}^{-1}\right)=1\) These are true. Again, \((\mathrm{A}+\mathrm{B})^{-1}=\frac{\operatorname{adj}(\mathrm{A}+\mathrm{B})}{|\mathrm{A}+\mathrm{B}|}\) \((\mathrm{A}+\mathrm{B})^{-1} \neq \mathrm{B}^{-1}+\mathrm{A}^{-1} \tag{iv}\)
Karnataka CET-2021
Matrix and Determinant
78780
If \(A=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]\), then \(A^{4}\) is equal to
78782
If \(\left[\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}2 \\ 4\end{array}\right]\), then the values of \(x\) and \(y\) respectively are
1 \(–3, –1\)
2 \(1, 3\)
3 \(3,1\)
4 \(-1,3\)
Explanation:
(D) : \({\left[\begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array}\right]_{2 \times 2}\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \end{array}\right]_{2 \times 1}=\left[\begin{array}{l} 2 \\ 4 \end{array}\right]}\) \({\left[\begin{array}{ccc} \mathrm{x} & +\mathrm{y} \\ -\mathrm{x} & +\mathrm{y} \end{array}\right]=\left[\begin{array}{l} 2 \\ 4 \end{array}\right]}\) On comparing corresponding elements on both the side we get - \(x+y=2 \tag{i}\) \(-x+y=4\) On adding equation (i) and (ii), we get - \(2 y=6 \Rightarrow y=3 \tag{ii}\) Putting in equation, we get - \(x+3=2\) \(x=-1\) The value of \(x\) and \(y\) respectively are \((-1,3)\)
(D) : According to question, A and B are invertible matrices it means - \((\mathrm{AB})^{-1}=\mathrm{B}^{-1} \mathrm{~A}^{-1} \tag{i}\) We know that, \(\mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}\) \(\operatorname{adj} \mathrm{A}=|\mathrm{A}| \mathrm{A}^{-1} \tag{ii}\) Also, \(\quad \operatorname{det}\left(\mathrm{A}^{-1}\right)=[\operatorname{det}(\mathrm{A})]^{-1}\) \(\operatorname{det}(A)^{-1}=\frac{1}{[\operatorname{det}(A)]} \tag{iii}\) \(\operatorname{det}(\mathrm{A}) \cdot \operatorname{det}\left(\mathrm{A}^{-1}\right)=1\) These are true. Again, \((\mathrm{A}+\mathrm{B})^{-1}=\frac{\operatorname{adj}(\mathrm{A}+\mathrm{B})}{|\mathrm{A}+\mathrm{B}|}\) \((\mathrm{A}+\mathrm{B})^{-1} \neq \mathrm{B}^{-1}+\mathrm{A}^{-1} \tag{iv}\)
Karnataka CET-2021
Matrix and Determinant
78780
If \(A=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]\), then \(A^{4}\) is equal to
78782
If \(\left[\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}2 \\ 4\end{array}\right]\), then the values of \(x\) and \(y\) respectively are
1 \(–3, –1\)
2 \(1, 3\)
3 \(3,1\)
4 \(-1,3\)
Explanation:
(D) : \({\left[\begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array}\right]_{2 \times 2}\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \end{array}\right]_{2 \times 1}=\left[\begin{array}{l} 2 \\ 4 \end{array}\right]}\) \({\left[\begin{array}{ccc} \mathrm{x} & +\mathrm{y} \\ -\mathrm{x} & +\mathrm{y} \end{array}\right]=\left[\begin{array}{l} 2 \\ 4 \end{array}\right]}\) On comparing corresponding elements on both the side we get - \(x+y=2 \tag{i}\) \(-x+y=4\) On adding equation (i) and (ii), we get - \(2 y=6 \Rightarrow y=3 \tag{ii}\) Putting in equation, we get - \(x+3=2\) \(x=-1\) The value of \(x\) and \(y\) respectively are \((-1,3)\)
(D) : According to question, A and B are invertible matrices it means - \((\mathrm{AB})^{-1}=\mathrm{B}^{-1} \mathrm{~A}^{-1} \tag{i}\) We know that, \(\mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}\) \(\operatorname{adj} \mathrm{A}=|\mathrm{A}| \mathrm{A}^{-1} \tag{ii}\) Also, \(\quad \operatorname{det}\left(\mathrm{A}^{-1}\right)=[\operatorname{det}(\mathrm{A})]^{-1}\) \(\operatorname{det}(A)^{-1}=\frac{1}{[\operatorname{det}(A)]} \tag{iii}\) \(\operatorname{det}(\mathrm{A}) \cdot \operatorname{det}\left(\mathrm{A}^{-1}\right)=1\) These are true. Again, \((\mathrm{A}+\mathrm{B})^{-1}=\frac{\operatorname{adj}(\mathrm{A}+\mathrm{B})}{|\mathrm{A}+\mathrm{B}|}\) \((\mathrm{A}+\mathrm{B})^{-1} \neq \mathrm{B}^{-1}+\mathrm{A}^{-1} \tag{iv}\)
Karnataka CET-2021
Matrix and Determinant
78780
If \(A=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]\), then \(A^{4}\) is equal to
78782
If \(\left[\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}2 \\ 4\end{array}\right]\), then the values of \(x\) and \(y\) respectively are
1 \(–3, –1\)
2 \(1, 3\)
3 \(3,1\)
4 \(-1,3\)
Explanation:
(D) : \({\left[\begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array}\right]_{2 \times 2}\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \end{array}\right]_{2 \times 1}=\left[\begin{array}{l} 2 \\ 4 \end{array}\right]}\) \({\left[\begin{array}{ccc} \mathrm{x} & +\mathrm{y} \\ -\mathrm{x} & +\mathrm{y} \end{array}\right]=\left[\begin{array}{l} 2 \\ 4 \end{array}\right]}\) On comparing corresponding elements on both the side we get - \(x+y=2 \tag{i}\) \(-x+y=4\) On adding equation (i) and (ii), we get - \(2 y=6 \Rightarrow y=3 \tag{ii}\) Putting in equation, we get - \(x+3=2\) \(x=-1\) The value of \(x\) and \(y\) respectively are \((-1,3)\)