91Ó°ÊÓ

Evaluate the determinants in Exercises 1 and 2 . (i) \(\left|\begin{array}{ll}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right|\) (ii) \(\left|\begin{array}{cc}x^{2}-x+1 & x-1 \\ x+1 & x+1\end{array}\right|\)

Examine the consistency of the system of equations in Exercises 1 to 6. $$ \begin{array}{l} x+3 y=5 \\ 2 x+6 y=8 \end{array} $$

Evaluate \(\left|\begin{array}{ccc}\cos \alpha \cos \beta & \cos \alpha \sin \beta & -\sin \alpha \\ -\sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{array}\right|\)

Verify \(\mathrm{A}(\operatorname{adj} \mathrm{A})=(a d j \mathrm{~A}) \mathrm{A}=|\mathrm{A}| \mathrm{I}\) in Exercises 3 and 4 $$ \begin{array}{cc} 2 & 3 \\ 4 & 6 \end{array} $$

Evaluate the determinants in Exercises 1 and 2 . If \(A=\left[\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right]\), then show that \(|2 \mathrm{~A}|=4|\mathrm{~A}|\)

Using the property of determinants and without expanding in Exercises 1 to 7 , prove that: $$ \left|\begin{array}{lll} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{array}\right|=0 $$

Find values of \(k\) if area of triangle is 4 sq. units and vertices are (i) \((k, 0),(4,0),(0,2)\) (ii) \((-2,0),(0,4),(0, k)\)

Using Cofactors of elements of second row, evaluate \(\Delta=\left|\begin{array}{lll}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{array}\right|\).

Verify \(\mathrm{A}(\operatorname{adj} \mathrm{A})=(a d j \mathrm{~A}) \mathrm{A}=|\mathrm{A}| \mathrm{I}\) in Exercises 3 and 4 $$ \begin{array}{ccc} 1 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 3 \end{array} $$

Using Cofactors of elements of third column, evaluate \(\Delta=\left|\begin{array}{lll}1 & x & y z \\ 1 & y & z x \\ 1 & z & x y\end{array}\right|\).