JEE Main B.Arch Qualifying Marks 2025 - Check Previous Year Cutoffs

Singular Matrix - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Determinant of a Matrix, Singular and Non-singular Matrix is considered one of the most asked concept.

  • 96 Questions around this concept.

Solve by difficulty

If

$\left|\begin{array}{ccc}a^2 & b^2 & c^2 \\ (a+\lambda)^2 & (b+\lambda)^2 & (c+\lambda)^2 \\ (a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2\end{array}\right|=k \lambda\left|\begin{array}{ccc}a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1\end{array}\right|, \lambda \neq 0$,

then k is equal to:

If $a>0$ and discriminant of $a x^2+2 b x+c$ is -ve, then
$
\left|\begin{array}{ccc}
a & b & a x+b \\
b & c & b x+c \\
a x+b & b x+c & 0
\end{array}\right|
$  is

If $1, \omega, \omega^2$ are the cube roots of unity, then $\Delta=\left|\begin{array}{ccc}1 & \omega^n & \omega^{2 n} \\ \omega^n & \omega^{2 n} & 1 \\ \omega^{2 n} & 1 & \omega^n\end{array}\right|$ is equal to

JEE Main 2025: Rank Predictor | College Predictor | Marks vs Rank vs Percentile

JEE Main 2025: Sample Papers | Syllabus | Mock Tests | PYQsHigh Scoring Topics

Apply to TOP B.Tech/BE Entrance exams: VITEEE | MET | AEEE | BITSAT

Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be such that $b(a+c) \neq 0$. If

$
\left|\begin{array}{ccc}
a & a+1 & a-1 \\
-b & b+1 & b-1 \\
c & c-1 & c+1
\end{array}\right|+\left|\begin{array}{ccc}
a+1 & b+1 & c-1 \\
a-1 & b-1 & c+1 \\
(-1)^{n+2} a & (-1)^{n+1} b & (-1)^n c
\end{array}\right|=0
$

then the value of $n$ is

$A=\left[\begin{array}{ccc}e^t & e^{-t} \cos t & e^{-t} \sin t \\ e^t & -e^{-t} \cos t-e^{-t} \sin t & -e^{-t} \sin t+e^{-t} \cos t \\ e^t & 2 e^{-t} \sin t & -2 e^{-t} \cos t\end{array}\right]$ then |A| is :

Find the value of a+b+c if  $\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]=0 \quad$ and $a \neq b \neq c \neq 0$

Find the value of x such that matrix A is singular, where $A=\left[\begin{array}{cc}1 & -x \\ 1 & x^4\end{array}\right]$

Amity University Noida B.Tech Admissions 2025

Among Top 30 National Universities for Engineering (NIRF 2024) | 30+ Specializations | AI Powered Learning & State-of-the-Art Facilities

Amrita Vishwa Vidyapeetham | B.Tech Admissions 2025

Recognized as Institute of Eminence by Govt. of India | NAAC ‘A++’ Grade | Upto 75% Scholarships

Which one of the true about the following determinant?

$\left[\begin{array}{ccc}\lim _{x \rightarrow 0} \frac{16^x-1}{x} & \sum_{r=1}^n(2 r-1) & \int \tan (x) d x \\ \lim _{x \rightarrow 0} \frac{\log (16 x+1)}{x} & \sum_{r=1}^n(2 r+1) & \int \cot (x) d x \\ 2 \log _e 4 & n^2 & -\log |\cos x|\end{array}\right]$

 

 

$
\begin{aligned}
&\text { If }\left|\begin{array}{ccc}
y+z & x & x \\
y & z+x & y \\
z & z & x+y
\end{array}\right|=K(x y z)\\
&\text { Then } \mathrm{K} \text { is equal to }
\end{aligned}
$

JEE Main 2025 College Predictor
Know your college admission chances in NITs, IIITs and CFTIs, many States/ Institutes based on your JEE Main result by using JEE Main 2025 College Predictor.
Try Now

$
\begin{aligned}
&\text { If w }(\neq 1) \text { is a cube root of unity then }\\
&\left|\begin{array}{ccc}
1 & 1+i+w^2 & w^2 \\
1-i & -1 & w^2-1 \\
-i & -i+w-1 & -1
\end{array}\right|=
\end{aligned}
$

Concepts Covered - 2

Determinant of a Matrix, Singular and Non-singular Matrix

The determinant of a matrix A is a number which is calculated from the matrix. For determinant to exist, matrix A must be a square matrix. Determinant of matrix is denoted by det A or |A|. 

For $2 \times 2$ matrices

$
\mathrm{A}=\left[\begin{array}{ll}
a_1 & a_2 \\
b_1 & b_2
\end{array}\right]
$

then $\operatorname{det} \mathrm{A}$ is :

$
|\mathrm{A}|=\left|\begin{array}{ll}
a_1 & a_2 \\
b_1 & b_2
\end{array}\right|=\mathrm{a}_1 \times \mathrm{b}_2-\mathrm{a}_2 \times \mathrm{b}_1
$
For a $3 \times 3$ matrix determinant can be calculated in the following way :
let $\mathrm{A}=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$
then we find $\operatorname{det} \mathrm{A}$ in following way

$
|A|=a_1\left(b_2 \cdot c_3-b_3 \cdot c_2\right)-a_2\left(b_1 \cdot c_3-c_1 b_3\right)+a_3\left(b_1 c_2-b_2 c_1\right)
$

This same process we follow to evaluate the determinant of the matrix of any order. Notice that we start first term with +ve sign and then 2nd with -ve sign and 3rd again +ve sign, this sign sequence is followed for any order of matrix.

This whole process is row dependent, same process can be done using column, means we can select element along column and delete their row and column and compute the determinant of left out matrix and then multiply it with the element which we select. And we will get the same result as we get while doing the whole process along row.

Singular and non-singular matrix:

A square matrix is called singular matrix if its determinant is 0 otherwise it is called non-singular matrix. Let say A is a square matrix then it is singular if |A| = 0, otherwise, it will be non-singular if |A| ≠ 0.

 

Properties of Determinant of a Matrix

Properties of Determinants - Part 4

If $A$ and $B$ are square matrices of same order:
i) $\operatorname{det}\left(A^{\prime}\right)=\operatorname{det} A$
ii) $\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)$ and $|A B|=|B A| \mid$
iii) if $A$ is skew symmetric matrix of odd order then $|A|=0$.
iv) if $A$ is a skew symmetric matrix of even order then $|A|$ is a perfect square.
v) $|k A|=k^n|A|$, where $n$ is order of $A$
vii) $\left|A^n\right|=|A|^n$, where $n$ belongs to$N$.

Study it with Videos

Determinant of a Matrix, Singular and Non-singular Matrix
Properties of Determinant of a Matrix

"Stay in the loop. Receive exam news, study resources, and expert advice!"

Get Answer to all your questions

Back to top