Using elementary row operations, find the inverse of the following matrix:

$\left[ \begin{array}{cc}2 & 5\\ 1 & 3\end{array} \right]$

For a 2 x 2 matrix, A = [ a_ij ] whose elements are given by  a_ij = $\frac{\mathrm{i}}{\mathrm{j}}$,  write the value of _a12 .

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For what value of x, the matrix $\left[ \begin{array}{cc}5 - \mathrm{x} & \mathrm{x} + 1\\ 2 & 4\end{array} \right]$ is singular?

Write  A^-1  for A = $\left[ \begin{array}{cc}2 & 5\\ 1 & 3\end{array} \right]$

Using elementary transformations, find the inverse of the matrx

$\left( \begin{array}{ccc}1 & 3& - 2\\ - 3 & 0 & - 1\\ 2& 1& 0\end{array} \right)$

If  $\left( \begin{array}{cc}2 & 3\\ 5 & 7\end{array} \right) \left(\begin{array}{cc}1& - 3\\ - 2 & 4\end{array} \right) = \left( \begin{array}{cc}- 4& 6\\ - 9& \mathrm{x}\end{array} \right)$,  write the value of x.

Simplify: $\cos \mathrm{\theta } \left[ \begin{array}{cc}\cos \mathrm{\theta } & \sin \mathrm{\theta }\\ - \sin \mathrm{\theta } & \cos \mathrm{\theta }\end{array} \right] + \sin \mathrm{\theta } \left[ \begin{array}{cc}\sin \mathrm{\theta } & - \cos \mathrm{\theta }\\ \cos \mathrm{\theta } & \sin \mathrm{\theta }\end{array} \right]$

Using elementary operations, find the inverse of the following matrix:

$\left(\begin{array}{ccc}- 1 & 1 & 2\\ 1& 2 & 3\\ 3& 1 & 1\end{array} \right)$

If P =  is the adjoint of a 3 x3 matrix A and |A| = 4, then α is equal to 

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Let A = . If u₁ and u₂ are column matrices such that Au1 =  and Au2 = , then u₁ +u₂ is equal to