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Class 9 Math Matrices and Determinants


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Class 9Class 10First YearSecond Year
1.Findtheinversewhereverpossibleofeachofthefollowingmatricesbytheadjointmethod.(ix)[1ωω2ωω21ω21ω]whereωisacomplexcuberootofunity.1. Find the inverse wherever possible of each of the following matrices by the adjoint method.(ix) \left[\begin{array}{lll}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\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=
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=

17.LetA=[130121002]B=[234123112]andC=[123358276]Showthat:(V)A+B=B+A7=±=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|
Evaluate the following determinants using their properties.8.  \left|\begin{array}{lll}x & y & z \\ z & x & y \\ y & z & x\end{array}\right|

Evaluatethefollowingdeterminantsusingtheirproperties.8.xyzzxyyzxEvaluate 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.
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.

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

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 .
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 .

8.LetA=[210143085]andX=[211324540]Find:(ii)λIAwhereλisanyscalarandIisaunitmatrixoforder3.8. 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|
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|

Evaluatethefollowingdeterminantsusingtheirproperties.7.a+b+2cbacbb+c+2acc+a+2baEvaluate 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|

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