Embibe Experts Solutions for Chapter: Matrices and Determinants, Exercise 1: NDA & NA EE 2019 Paper 1 Mathematics

For how many values of $k,$ is the matrix $\left[\begin{array}{rrr}0& k& 4\\ -k& 0& -5\\ -k& k& -1\end{array}\right]$ singular?

Let $A=\left[\begin{array}{cc}x+y& y\\ 2x& x-y\end{array}\right],B=\left[\begin{array}{c}2\\ -1\end{array}\right]$and $C=\left[\begin{array}{l}3\\ 2\end{array}\right] .$ If $AB=C,$ then what is the value of the determinant of the
matrix $A ?$

What is the value of the determinant $\left|\begin{array}{ccc}i& i^2& i^3\\ i^4& i^6& i^8\\ i^9& i^{12}& i^{15}\end{array}\right|$ where $i=\sqrt{-1}?$

Let $A=\left[\begin{array}{lll}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$ and $B=\left[\begin{array}{l}x\\ y\\ z\end{array}\right]$ then what is $AB$ equal to?

If $A$ is a matrix of order $3\times 5$ and $B$ is a matrix of order $5\times 3,$ then the order of $AB$ and $BA$ will respectively be

Let $p, q$ and $r$ be three distinct positive real numbers. If $D=\left|\begin{array}{lll}p& q& r\\ q& r& p\\ r& p& q\end{array}\right|,$ then which one of the following is correct?

Consider the following in respect of a non-singular matrix of order $3.$

$1$. $A(adjA)=(adjA)A$

$2$. $|adjA|=\left|A\right|$

Which of the above statements is/are correct?

Under which condition, are the points $(a, b), (c, d)$ and $(a-c, b-d)$ collinear?