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##### Evaluate the following determinants using their properties.7. \left|\begin{array}{ccc}a+b+2 c & b & a \\ c & b & b+c+2 a \\ c & c+a+2 b & a\end{array}\right|

Q.20 If A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] then adj A equals :(a) \left[\begin{array}{rr}d & b \\ -c & a\end{array}\right] (b) \left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right] (c) \left[\begin{array}{ll}d & b \\ c & a\end{array}\right] (d) \left[\begin{array}{rr}d & -b \\ c & a\end{array}\right] Gujranwala Board 2007

Evaluate the following determinants using their properties.8. \left|\begin{array}{lll}x & y & z \\ z & x & y \\ y & z & x\end{array}\right|

Evaluate the following determinants:6. \left|\begin{array}{lll}1 & 1 & \omega \\ 1 & 1 & \omega^{2} \\ \omega & \omega^{2} & 1\end{array}\right| where \omega is a complex cube root of unity.

17. Let \mathrm{A}=\left[\begin{array}{ccc}1 & 3 & 0 \\ -1 & 2 & 1 \\ 0 & 0 & 2\end{array}\right] \mathrm{B}=\left[\begin{array}{ccc}2 & 3 & 4 \\ 1 & 2 & 3 \\ -1 & 1 & 2\end{array}\right] and C=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 5 & 8 \\ 2 & 7 & 6\end{array}\right] Show that :(V) \mathrm{A}+\mathrm{B}=\mathrm{B}+\mathrm{A}^{-7}=\pm=

(vi) Product of \left[\begin{array}{ll}x & y\end{array}\right]\left[\begin{array}{c}2 \\ -1\end{array}\right] is(a) [2 x+y] (b) [x-2 y] (c) [2 x-y] (d) [x+2 y]

Q.14 \left[\begin{array}{lll}0 & 0 & 0\end{array}\right] is(a) scalar matrix(b) diagonal matrix(c) identity matrix(d) null matrix