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Mathematics
Matrices and determinants
Obtain the inverse of the following matrix using elementary operations $A = \left[ {\begin{array}{*{20}{c}}
  0&1&2 \\
  1&2&3 \\
  3&1&1
\end{array}} \right]$

Mathematics
Matrices and determinants
Find x and y, if \[x + y = \left[ {\begin{array}{*{20}{c}}
  7&0 \\
  2&5
\end{array}} \right]\] and \[x - y = \left[ {\begin{array}{*{20}{c}}
  3&0 \\
  0&3
\end{array}} \right]\].

Mathematics
Matrices and determinants
Write the value $\left| \begin{matrix}
   x+y & y+z & z+x \\
   z & x & y \\
   -3 & -3 & -3 \\
\end{matrix} \right|$

Mathematics
Matrices and determinants
If \[A = \left[ {\begin{array}{*{20}{c}}
  1&2&2 \\
  2&1&2 \\
  2&2&1
\end{array}} \right]\] , then prove that ${A^2} - 4A - 5I = 0$ , also find ${A^{ - 1}}$ .

Mathematics
Matrices and determinants
If ${A^3} = O$, then $I + A + {A^2}$ equals
A.$I - A$
B.$\left( {I + {A^{ - 1}}} \right)$
C.${\left( {I - A} \right)^{ - 1}}$
D.None of these

Mathematics
Matrices and determinants
Write the value of x if \[\left|\begin{align}
  & 2x\text{ 5} \\
 & \text{8 x} \\
\end{align} \right|\] =\[\left|\begin{align}
  & \text{6 -2} \\
 & \text{7 3} \\
\end{align} \right|\] .
Mathematics
Matrices and determinants
Without expanding the determinant, prove that \[\left. \left| \begin{matrix}
   a & {{a}^{2}} & bc \\
   b & {{b}^{2}} & ca \\
   c & {{c}^{2}} & ab \\
\end{matrix} \right. \right|=\left. \left| \begin{matrix}
   1 & {{a}^{2}} & {{a}^{3}} \\
   1 & {{b}^{2}} & {{b}^{3}} \\
   1 & {{c}^{2}} & {{c}^{3}} \\
\end{matrix} \right. \right|\].
Mathematics
Matrices and determinants
If $\alpha ,\beta \text{ and }\gamma $ are the roots of the equation ${{x}^{3}}+px+q=0$ then the value of determinant \[\left| \begin{matrix}
   \alpha & \beta & \gamma \\
   \beta & \gamma & \alpha \\
   \gamma & \alpha & \beta \\
\end{matrix} \right|\] is
\[\begin{align}
  & A.p \\
 & B.q \\
 & C.{{p}^{2}}-2q \\
 & D.0 \\
\end{align}\]

Mathematics
Matrices and determinants
Let M be a 2 x 2 symmetric matrix with integer entries. Then M is invertible if \[\]
A. The first column of M is the transpose of the second row of M\[\]
B. The second row of M is the transpose of the first column of M \[\]
C. M is a diagonal matrix with non-zero entries in the main diagonal\[\]
D. The product of entries in the main diagonal of M is not the square of an integer\[\]
Mathematics
Matrices and determinants
Using properties of determinants, show that \[\left| \begin{matrix}
   a+b & a & b \\
   a & a+c & c \\
   b & c & b+c \\
\end{matrix} \right|=4abc\]

Mathematics
Matrices and determinants
If \[\alpha \] , \[\beta \ne 0\] , and \[f\left( n \right) = {\alpha ^n} + {\beta ^n}\] and \[\left| {\begin{array}{*{20}{c}}
  3&{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)} \\
  {1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)} \\
  {1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}&{1 + f\left( 4 \right)}
\end{array}} \right| = K{\left( {1 - \alpha } \right)^2}{\left( {1 - \beta } \right)^2}{\left( {\alpha - \beta } \right)^2}\] , then K is equal to
A \[\alpha \beta \]
B \[\dfrac{1}{{\alpha \beta }}\]
C 1
D -1

Mathematics
Matrices and determinants
If $A$ and $B$ are square matrices of order $3$ such that $\left| A \right|=-1$, $\left| B \right|=3$, then $\left| 3AB \right|$ is equal to
A. $-9$
B. $-81$
C. $-27$
D. 81
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