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##### Q.58 If \left|\begin{array}{cc}-1 & 3 \\ x & 1\end{array}\right|=0 then x= (a) 3(b) -3 (c) \frac{1}{3} (d) -\frac{1}{3} Lahore Board 2015; Multan Board 2004

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=

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

18. Prove the following identities :$\left\{\left[\begin{array}{lll}1 & \omega & \omega^{2} \\\omega & \omega^{2} & 1 \\\omega^{2} & 1 & \omega\end{array}\right]+\left[\begin{array}{lll}\omega & \omega^{2} & 1 \\\omega^{2} & 1 & \omega \\\omega & \omega^{2} & 1\end{array}\right]\right\}\left[\begin{array}{l}1 \\\omega \\\omega^{2}\end{array}\right]=\left[\begin{array}{l}0 \\0 \\0\end{array}\right]$where \omega is a complex cube root of unity.

1. Use matrices if possible to solve the following systems of linear equations by:(i) the matrix inversion method (ii) the Cramers rule.(vii)$\begin{array}{l}2 x-2 y=4 \\-5 x-2 y=-10\end{array}$

8. Let A=\left[\begin{array}{ccc}-2 & 1 & 0 \\ -1 & 4 & 3 \\ 0 & 8 & 5\end{array}\right] and X=\left[\begin{array}{ccc}2 & 1 & -1 \\ -3 & 2 & -4 \\ 5 & 4 & 0\end{array}\right] Find:(ii) \lambda I-A where \lambda is any scalar and I is a unit matrix of order 3 .

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|