Question

In the interval ${\frac{\pi}{2}}<x<\pi$ , find the value of x for which that matrix $\left(\begin{array}{l l }2sinx & 3 \\ 1 & 2sinx\end{array}\right)$ is singular.

If A and B are invertible matrices of order 3, $|\mathbf{A}|=$ 2 and $|(\mathbf{AB})^{-1}|\,=\,-{\frac{1}{6}}.$ Find $|\mathbf{B}$.

If A = $\left(\begin{array}{l l }4 & 6 \\ 7 & 5\end{array}\right)$ , then what is the value of $\ A{\cdot}(a d j A)$ ?

Find the solution of the differential equation ${\frac{d y}{d x}}=x^{3}e^{-2y}$.

Write the projection of the vector $\stackrel{\rightarrow}{a}\$ = $2\stackrel{\wedge}{i}-\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ on the vector $\stackrel{\rightarrow}{b}\$ = $\stackrel{\wedge}{i}+2\stackrel{\wedge}{j} +2\stackrel{\wedge}{k}$.

If $\begin{array}{l l}{{\vec{a}=x\,\hat{i}+2\,\hat{j}-z\,\hat{k}~}}&{{\mathrm{and}~~~~\vec{b}=3\,\hat{i}-y\,\hat{j}+\hat{k}}}\end{array}$ are two equal vectors, then write the value of x + y + z.

If A is a 3 × 3 matrix, whose elements are given by $a_{i j}={\frac{1}{3}}|-3i+j\,|$ , then write the value of $a_{23}$.