Embibe Experts Solutions for Chapter: Matrices and Determinants, Exercise 1: WBJEE 2018

The least positive integer $n$ such that ${\left(\begin{array}{cc}\cos \frac{\pi }{4}& \sin \frac{\pi }{4}\\ -\sin \frac{\pi }{4}& \cos \frac{\pi }{4}\end{array}\right)}^n$ is an identity matrix of order $2$ is

In a third order matrix $A, a_{ij}$ denotes the element in the $i^{\mathrm{th}}$ row and $j^{\mathrm{th}}$ column.

If $a_{ij}=0$ for $i=j$
        $=1$ for $i>j$
        $=-1$ for $i<j$

Then the matrix is

Let the matrix, $A=\left(\begin{array}{ccc}3-t& 1& 0\\ -1& 3-t& 1\\ 0& -1& 0\end{array}\right)$ and det$A=5,$ then

Let $A=\left(\begin{array}{lll}3& 0& 3\\ 0& 3& 0\\ 3& 0& 3\end{array}\right).$ Then, the roots of the equation $det\left(A-\lambda I_3\right)=0 ($where $I_3$ is the identity matrix of order $3 )$ are

If the polynomial $f\left(x\right)=\left|\begin{array}{ccc}(1+x{)}^a& (2+x{)}^b& 1\\ 1& (1+x{)}^a& (2+x{)}^b\\ (2+x{)}^b& 1& (1+x{)}^a\end{array}\right|$ then the constant term of $f\left(x\right)$ is

$[a$ and $b$ are positive integers $]$

If $\left|\begin{array}{ccc}{a}^2& bc& c^2+ac\\ a^2+ab& b^2& ca\\ ab& b^2+bc& c^2\end{array}\right|=k{a}^2{b}^2{c}^2,$ then $k=$

Let $A=\left(\begin{array}{ccc}12& 24& 5\\ x& 6& 2\\ -1& -2& 3\end{array}\right).$ The value of $x$ for which the matrix $A$ is not invertible is

The system of equations

$\lambda x+y+3z=0$

$2x+\mu y-z=0$

$5x+7y+z=0$

has infinitely many solutions in $R.$ Then,