MULTIPLE CHOICE QUESTIONS (MCQs)

If $A$ is a square matrix satisfying $A^{T}A=I$, write the value of $\left|A\right|$.

If $A$ and $B$ are square matrices of the same order such that $\left|A\right|=3$ and $AB=I$, then write the value of $\left|B\right|$.

Write the value of the determinant $\left|\begin{array}{rrr}2& -3& 5\\ 4& -6& 10\\ 6& -9& 15\end{array}\right|$.

If the matrix $\left[\begin{array}{rr}5x& 2\\ -10& 1\end{array}\right]$ is singular, find the value of $x$.

If $A=\left[\begin{array}{ccc}5& 3& 8\\ 2& 0& 1\\ 1& 2& 3\end{array}\right]$. Write the cofactor of the element $a_{32}$.

If $\left|\begin{array}{cc}x+1& x-1\\ x-3& x+2\end{array}\right|=\left|\begin{array}{cc}4& -1\\ 1& 3\end{array}\right|$, then write the value of $x$.

Find the maximum value of $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+\sin \theta & 1\\ 1& 1& 1+\cos \theta \end{array}\right|$

If $\left|\begin{array}{ccc}x& \sin \theta & \cos \theta \\ -\sin \theta & -x& 1\\ \cos \theta & 1& x\end{array}\right|=8$, write the value of $x$.