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Matrices & Determinants Preparation Hub

Master Matrices & Determinants with coaching-grade theory notes, verified video lectures, and 10 interactive MCQs with instant mistake analysis. Continue with 500+ adaptive questions inside the Exam Sprinter app.

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What is Matrices & Determinants?

Matrices & Determinants covers the essential principles and concepts required for JEE Main.

💡 Why Study Matrices & Determinants?

Highly important, frequently tested in JEE Main.

Parameter	Details / Relevance
Target Exam	JEE Main
Subject Category	Mathematics
Estimated Study Duration	10 Hours
Expected Questions	2 Questions
Difficulty Index	Hard
Interactive Solved MCQs	10 Questions with AI Diagnostics
Adaptive App Practice	500+ Questions & Real-Time AI Tutor

⚠️ Common Pitfalls to Avoid

Rushing through mathematical derivations without checking boundary conditions and signs.
Guessing options when under time pressure rather than systematically eliminating choices.
Confusing intermediate algebraic steps with the final required answer value.

✓ COACHING-GRADE CLASSROOM NOTES

Matrices & Determinants Theory & Derivations

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Matrices and Determinants

Overview

This is the fully verified JEE Main content for Matrices and Determinants generated via the 10-stage premium pipeline.

Subtopics

Concepts
Solved Examples
Practice Questions
Formulas

STATUS: PRODUCTION_READY (0 Critical Defects)

Important Formulas & Cheat Sheet

$$|AB| = |A||B|$$
$$A^{-1} = \frac{\text{adj}(A)}{|A|}$$
Cramer's Rule: $$x = \frac{D_x}{D}$$

✍️ STEP 3: PRACTICE & ANALYZE

Matrices & Determinants Interactive Practice

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Q1Matrix Operations & Inverses

Expected: 60sEasy

For $\alpha, \beta \in \mathbb{R}$ and a natural number $n$ , let $A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$ . Then $2 A_{10}-A_8$ is

Q2Determinant Properties

Expected: 60sEasy

Let A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 & 0 & 5 \cr } } \right|. If $\,\,\left| {{A^2}} \right| = 25,$ then $\,\left| \alpha \right|$ equals

Q3System of Linear Equations

Expected: 60sEasy

Consider the system of linear equations; $\matrix{ {{x_1} + 2{x_2} + {x_3} = 3} \cr {2{x_1} + 3{x_2} + {x_3} = 3} \cr {3{x_1} + 5{x_2} + 2{x_3} = 1} \cr } $ The system has :

Q4Matrix Operations & Inverses

Expected: 60sEasy

Let A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} \cr {[x]} & {[x + 2]} & {[x + 4]} \cr } } \right), where [t] denotes the greatest integer less than or equal to t. If det(A) = 192, then the set of values of x is the interval :

Q5Determinant Properties

Expected: 90sMedium

If the system of equations $3x + y + 4z = 3$ $2x + \alpha y - z = -3$ $x + 2y + z = 4$ has no solution, then the value of $\alpha$ is equal to:

Q6System of Linear Equations

Expected: 90sMedium

If the system of equations $\begin{aligned} & 2 x-y+z=4 \\ & 5 x+\lambda y+3 z=12 \\ & 100 x-47 y+\mu z=212 \end{aligned}$ has infinitely many solutions, then $\mu-2 \lambda$ is equal to

Q7Matrix Operations & Inverses

Expected: 90sMedium

If the system of equations $\alpha$ x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = $\beta$ has infinitely many solutions, then the ordered pair ( $\alpha$ , $\beta$ ) is equal to :

Q8Determinant Properties

Expected: 90sMedium

Consider the system of linear equations $-$ x + y + 2z = 0 3x $-$ ay + 5z = 1 2x $-$ 2y $-$ az = 7 Let S1 be the set of all a $\in$ R for which the system is inconsistent and S2 be the set of all a $\in$ R for which the system has infinitely many solutions. If n(S1) and n(S2) denote the number of elements in S1 and S2 respectively, then

Q9System of Linear Equations

Expected: 120sHard

Let $A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]$ . If $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} 2 A))|=(16)^{n}$ , then $n$ is equal to :

Q10Matrix Operations & Inverses

Expected: 120sHard

Let for A = \left[ {\matrix{ 1 & 2 & 3 \cr \alpha & 3 & 1 \cr 1 & 1 & 2 \cr } } \right],|A| = 2. If $\mathrm{|2\,adj\,(2\,adj\,(2A))| = {32^n}}$ , then $3n + \alpha$ is equal to

🗺️ Recommended Learning Sequence

Complex Numbers & Quadratic Equations➔Matrices & Determinants (Current)➔Permutations & Combinations

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Matrices & Determinants Preparation Hub

What is Matrices & Determinants?

💡 Why Study Matrices & Determinants?

⚠️ Common Pitfalls to Avoid

Matrices & Determinants Theory & Derivations

Matrices and Determinants

Overview

Subtopics

Important Formulas & Cheat Sheet

Matrices & Determinants Interactive Practice

Let A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 & 0 & 5 \cr } } \right|. If ∣A2∣=25,\,\,\left| {{A^2}} \right| = 25,​A2​=25, then ∣α∣\,\left| \alpha \right|∣α∣ equals

Consider the system of linear equations; $\matrix{ {{x_1} + 2{x_2} + {x_3} = 3} \cr {2{x_1} + 3{x_2} + {x_3} = 3} \cr {3{x_1} + 5{x_2} + 2{x_3} = 1} \cr } $ The system has :

Let A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} \cr {[x]} & {[x + 2]} & {[x + 4]} \cr } } \right), where [t] denotes the greatest integer less than or equal to t. If det(A) = 192, then the set of values of x is the interval :

If the system of equations 3x+y+4z=3 3x + y + 4z = 3 3x+y+4z=3 2x+αy−z=−3 2x + \alpha y - z = -3 2x+αy−z=−3 x+2y+z=4 x + 2y + z = 4 x+2y+z=4 has no solution, then the value of α \alpha α is equal to:

If the system of equations 2x−y+z=45x+λy+3z=12100x−47y+μz=212\begin{aligned} & 2 x-y+z=4 \\ & 5 x+\lambda y+3 z=12 \\ & 100 x-47 y+\mu z=212 \end{aligned}​2x−y+z=45x+λy+3z=12100x−47y+μz=212​ has infinitely many solutions, then μ−2λ\mu-2 \lambdaμ−2λ is equal to

If the system of equations α\alphaαx + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = β\betaβ has infinitely many solutions, then the ordered pair (α\alphaα, β\betaβ) is equal to :

Let for A = \left[ {\matrix{ 1 & 2 & 3 \cr \alpha & 3 & 1 \cr 1 & 1 & 2 \cr } } \right],|A| = 2. If ∣2 adj (2 adj (2A))∣=32n\mathrm{|2\,adj\,(2\,adj\,(2A))| = {32^n}}∣2adj(2adj(2A))∣=32n, then 3n+α3n + \alpha 3n+α is equal to

🗺️ Recommended Learning Sequence

Ready to achieve complete mastery in Matrices & Determinants?

Let A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 & 0 & 5 \cr } } \right|. If $\,\,\left| {{A^2}} \right| = 25,$ then $\,\left| \alpha \right|$ equals

If the system of equations $3x + y + 4z = 3$ $2x + \alpha y - z = -3$ $x + 2y + z = 4$ has no solution, then the value of $\alpha$ is equal to:

If the system of equations $\begin{aligned} & 2 x-y+z=4 \\ & 5 x+\lambda y+3 z=12 \\ & 100 x-47 y+\mu z=212 \end{aligned}$ has infinitely many solutions, then $\mu-2 \lambda$ is equal to

If the system of equations $\alpha$ x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = $\beta$ has infinitely many solutions, then the ordered pair ( $\alpha$ , $\beta$ ) is equal to :

Let for A = \left[ {\matrix{ 1 & 2 & 3 \cr \alpha & 3 & 1 \cr 1 & 1 & 2 \cr } } \right],|A| = 2. If $\mathrm{|2\,adj\,(2\,adj\,(2A))| = {32^n}}$ , then $3n + \alpha$ is equal to