Question

If x ∈ N and $\begin{vmatrix}x+3 & -2 \\ -3x & 2x\end{vmatrix}$ = 8, then find the value of x.

Write the projection of the vector $\stackrel{\wedge}{i}+\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ along the vector $\stackrel{\rightarrow}{j}\$.

If $\begin{vmatrix} 2x & x+3 \\ 2(x+1) & x+1 \end{vmatrix} = \begin{vmatrix} 1 & 5 \\ 3 & 3 \end{vmatrix}$, write the value of x.

If matrix A = $\left(\begin{array}{l l }3 & -3 \\ -3 & 3\end{array}\right)$ and A² = λA, then write the value of λ.

If f is an invertible function, defined as f(x) = $\frac{3x-4}{5}$, then write $f^{-1}(x).$

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 square matrix such that A² = A, then write the value of (I + A)² – 3A.