(C) : If \(A\) is a square matrix of order \(n \times n\) \(\therefore \mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}\) \(\mathrm{A}^{-1}=\frac{\mathrm{I}}{\mathrm{A}}\) From equation (i) and (ii) we get \(\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}=\frac{\mathrm{I}}{\mathrm{A}}\) Aadj \(\mathrm{A}=|\mathrm{A}|\) Now, \(\operatorname{adj}(\operatorname{adj} A)=\) \(\operatorname{adJA} \cdot \operatorname{adj}(\operatorname{adj} A)=|\operatorname{adj} A|\) \(\{\because\) Re place A by city A in equation (iii) \(\}\) \(\operatorname{adj}(\operatorname{adj} A)=\frac{|\operatorname{adj} A|}{\operatorname{adj} A}\) from equation (iii), \(\operatorname{adj} \mathrm{A}=\frac{|\mathrm{A}|}{\mathrm{A}}\) \(\operatorname{adj}(\operatorname{adj} A)=\frac{|\operatorname{adj} A| A}{|A|} \Rightarrow \operatorname{adj}(\operatorname{adj} A)=\frac{|A|^{n-1} A}{|A|}\) \(\left\{\because|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{\mathrm{n}-1}\right\} \quad=|\mathrm{A}|^{\mathrm{n}-2} \mathrm{~A}\)
AMU-2014
Matrix and Determinant
78863
If \(A=\left[\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}\right]\) be such that \(A^{-1}=k A\), then \(k\) is equal to
78866
If \(k\) is one of the roots of the equation \(x^{2}-25 x+\) \(24=0\) such that \(A=\left[\begin{array}{lll}1 & 2 & 1 \\ 3 & 2 & 3\end{array}\right]\) is a nonsingular matrix, then \(A^{-1}=\left[\begin{array}{lll}1 & 1 & k\end{array}\right]\)
(C) : If \(A\) is a square matrix of order \(n \times n\) \(\therefore \mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}\) \(\mathrm{A}^{-1}=\frac{\mathrm{I}}{\mathrm{A}}\) From equation (i) and (ii) we get \(\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}=\frac{\mathrm{I}}{\mathrm{A}}\) Aadj \(\mathrm{A}=|\mathrm{A}|\) Now, \(\operatorname{adj}(\operatorname{adj} A)=\) \(\operatorname{adJA} \cdot \operatorname{adj}(\operatorname{adj} A)=|\operatorname{adj} A|\) \(\{\because\) Re place A by city A in equation (iii) \(\}\) \(\operatorname{adj}(\operatorname{adj} A)=\frac{|\operatorname{adj} A|}{\operatorname{adj} A}\) from equation (iii), \(\operatorname{adj} \mathrm{A}=\frac{|\mathrm{A}|}{\mathrm{A}}\) \(\operatorname{adj}(\operatorname{adj} A)=\frac{|\operatorname{adj} A| A}{|A|} \Rightarrow \operatorname{adj}(\operatorname{adj} A)=\frac{|A|^{n-1} A}{|A|}\) \(\left\{\because|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{\mathrm{n}-1}\right\} \quad=|\mathrm{A}|^{\mathrm{n}-2} \mathrm{~A}\)
AMU-2014
Matrix and Determinant
78863
If \(A=\left[\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}\right]\) be such that \(A^{-1}=k A\), then \(k\) is equal to
78866
If \(k\) is one of the roots of the equation \(x^{2}-25 x+\) \(24=0\) such that \(A=\left[\begin{array}{lll}1 & 2 & 1 \\ 3 & 2 & 3\end{array}\right]\) is a nonsingular matrix, then \(A^{-1}=\left[\begin{array}{lll}1 & 1 & k\end{array}\right]\)
(C) : If \(A\) is a square matrix of order \(n \times n\) \(\therefore \mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}\) \(\mathrm{A}^{-1}=\frac{\mathrm{I}}{\mathrm{A}}\) From equation (i) and (ii) we get \(\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}=\frac{\mathrm{I}}{\mathrm{A}}\) Aadj \(\mathrm{A}=|\mathrm{A}|\) Now, \(\operatorname{adj}(\operatorname{adj} A)=\) \(\operatorname{adJA} \cdot \operatorname{adj}(\operatorname{adj} A)=|\operatorname{adj} A|\) \(\{\because\) Re place A by city A in equation (iii) \(\}\) \(\operatorname{adj}(\operatorname{adj} A)=\frac{|\operatorname{adj} A|}{\operatorname{adj} A}\) from equation (iii), \(\operatorname{adj} \mathrm{A}=\frac{|\mathrm{A}|}{\mathrm{A}}\) \(\operatorname{adj}(\operatorname{adj} A)=\frac{|\operatorname{adj} A| A}{|A|} \Rightarrow \operatorname{adj}(\operatorname{adj} A)=\frac{|A|^{n-1} A}{|A|}\) \(\left\{\because|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{\mathrm{n}-1}\right\} \quad=|\mathrm{A}|^{\mathrm{n}-2} \mathrm{~A}\)
AMU-2014
Matrix and Determinant
78863
If \(A=\left[\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}\right]\) be such that \(A^{-1}=k A\), then \(k\) is equal to
78866
If \(k\) is one of the roots of the equation \(x^{2}-25 x+\) \(24=0\) such that \(A=\left[\begin{array}{lll}1 & 2 & 1 \\ 3 & 2 & 3\end{array}\right]\) is a nonsingular matrix, then \(A^{-1}=\left[\begin{array}{lll}1 & 1 & k\end{array}\right]\)
(C) : If \(A\) is a square matrix of order \(n \times n\) \(\therefore \mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}\) \(\mathrm{A}^{-1}=\frac{\mathrm{I}}{\mathrm{A}}\) From equation (i) and (ii) we get \(\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}=\frac{\mathrm{I}}{\mathrm{A}}\) Aadj \(\mathrm{A}=|\mathrm{A}|\) Now, \(\operatorname{adj}(\operatorname{adj} A)=\) \(\operatorname{adJA} \cdot \operatorname{adj}(\operatorname{adj} A)=|\operatorname{adj} A|\) \(\{\because\) Re place A by city A in equation (iii) \(\}\) \(\operatorname{adj}(\operatorname{adj} A)=\frac{|\operatorname{adj} A|}{\operatorname{adj} A}\) from equation (iii), \(\operatorname{adj} \mathrm{A}=\frac{|\mathrm{A}|}{\mathrm{A}}\) \(\operatorname{adj}(\operatorname{adj} A)=\frac{|\operatorname{adj} A| A}{|A|} \Rightarrow \operatorname{adj}(\operatorname{adj} A)=\frac{|A|^{n-1} A}{|A|}\) \(\left\{\because|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{\mathrm{n}-1}\right\} \quad=|\mathrm{A}|^{\mathrm{n}-2} \mathrm{~A}\)
AMU-2014
Matrix and Determinant
78863
If \(A=\left[\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}\right]\) be such that \(A^{-1}=k A\), then \(k\) is equal to
78866
If \(k\) is one of the roots of the equation \(x^{2}-25 x+\) \(24=0\) such that \(A=\left[\begin{array}{lll}1 & 2 & 1 \\ 3 & 2 & 3\end{array}\right]\) is a nonsingular matrix, then \(A^{-1}=\left[\begin{array}{lll}1 & 1 & k\end{array}\right]\)
(C) : If \(A\) is a square matrix of order \(n \times n\) \(\therefore \mathrm{A}^{-1}=\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}\) \(\mathrm{A}^{-1}=\frac{\mathrm{I}}{\mathrm{A}}\) From equation (i) and (ii) we get \(\frac{\operatorname{adj} \mathrm{A}}{|\mathrm{A}|}=\frac{\mathrm{I}}{\mathrm{A}}\) Aadj \(\mathrm{A}=|\mathrm{A}|\) Now, \(\operatorname{adj}(\operatorname{adj} A)=\) \(\operatorname{adJA} \cdot \operatorname{adj}(\operatorname{adj} A)=|\operatorname{adj} A|\) \(\{\because\) Re place A by city A in equation (iii) \(\}\) \(\operatorname{adj}(\operatorname{adj} A)=\frac{|\operatorname{adj} A|}{\operatorname{adj} A}\) from equation (iii), \(\operatorname{adj} \mathrm{A}=\frac{|\mathrm{A}|}{\mathrm{A}}\) \(\operatorname{adj}(\operatorname{adj} A)=\frac{|\operatorname{adj} A| A}{|A|} \Rightarrow \operatorname{adj}(\operatorname{adj} A)=\frac{|A|^{n-1} A}{|A|}\) \(\left\{\because|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{\mathrm{n}-1}\right\} \quad=|\mathrm{A}|^{\mathrm{n}-2} \mathrm{~A}\)
AMU-2014
Matrix and Determinant
78863
If \(A=\left[\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}\right]\) be such that \(A^{-1}=k A\), then \(k\) is equal to
78866
If \(k\) is one of the roots of the equation \(x^{2}-25 x+\) \(24=0\) such that \(A=\left[\begin{array}{lll}1 & 2 & 1 \\ 3 & 2 & 3\end{array}\right]\) is a nonsingular matrix, then \(A^{-1}=\left[\begin{array}{lll}1 & 1 & k\end{array}\right]\)
