Technique

Cofactor expansion

\det A = \sum_{j} a_{ij}C_{ij} \quad (\text{any fixed row } i)

$a_{ij}$: the entry in row i, column j
$C_{ij}$: its cofactor, the signed minor

A determinant equals the sum of the products of the entries of any one row or column with their own cofactors.

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#1NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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Set · 1 question
** Consider the determinant $f(x) = \left| \begin{matrix} x^{2} - 5x + 3 & 2x - 5 & 3 \\ 3x^{2} + x + 4 & 6x + 1 & 9 \\ 7x^{2} - 6x + 9 & 14x - 6 & 21 \end{matrix} \right|$
- #2NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#3NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
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#4NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#5NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#6NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
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Set · 1 question
Consider the following for the next three (03) items that follow: Let $\Delta$ be the determinant of a matrix A, where $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$ and $C_{11}, C_{12}, C_{13}$ be the cofactors of $a_{11}, a_{12}, a_{13}$ respectively.
- #7NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
  Strategy Concept notes
#8CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
#9CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
#10CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
#11NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#12NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#13NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#14NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#15NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
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#16NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#17NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#18NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#19NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#20NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
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#21NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#22NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#23NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#24NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#25NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#26NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#27NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#28NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#29NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#30NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
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#31NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#32NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#33NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#34NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#35NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#36CBSE Class 12 → Mathematics → Determinants → Minors and Cofactors
#37NDA → Mathematics → Matrices & Determinants → Linear Systems — Consistency, Cramer's Rule, Solution Space
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#38NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#39NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#40NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#41NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
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#42CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
#43NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#44CBSE Class 12 → Mathematics → Determinants → Adjoint and Inverse of a Matrix
#45CBSE Class 12 → Mathematics → Determinants → Adjoint and Inverse of a Matrix
#46CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
#47CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
#48MHT-CET → Maths → Determinants and Matrices → Determinants, Cofactors and the Adjoint Identities
Strategy Concept notes
Set · 1 question
Let $A=\begin{pmatrix}3&-3&4\\2&-3&4\\0&-1&1\end{pmatrix}$ .
- #49NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
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#50CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
#51NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
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#52CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
#53MHT-CET → Maths → Determinants and Matrices → Determinants, Cofactors and the Adjoint Identities
Strategy Concept notes
#54CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
#55NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#56NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#57NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#58NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#59NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#60NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#61NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#62NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#63NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#64NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#65NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#66NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#67NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
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#68NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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#69NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#70NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#71NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#72NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#73CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
#74NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#75NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
Strategy Concept notes
#76NDA → Mathematics → Matrices & Determinants → Linear Systems — Consistency, Cramer's Rule, Solution Space
Strategy Concept notes
#77NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
Strategy Concept notes
#78NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
Strategy Concept notes
#79NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#80NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#81NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
Strategy Concept notes
#82NDA → Mathematics → Matrices & Determinants → Linear Systems — Consistency, Cramer's Rule, Solution Space
Strategy Concept notes
#83NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
Strategy Concept notes
#84NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
Strategy Concept notes
#85NDA → Mathematics → Matrices & Determinants → Cofactors, Adjoint, and Inverse
Strategy Concept notes
#86MHT-CET → Maths → Determinants and Matrices → Determinants, Cofactors and the Adjoint Identities
Strategy Concept notes
#87NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
Strategy Concept notes
Set · 1 question
Consider the following for the next two (02) items that follow: Let $\Delta(a,b,c,\alpha)=\begin{vmatrix}a&b&a\alpha+b\\b&c&b\alpha+c\\a\alpha+b&b\alpha+c&0\end{vmatrix}$ .
- #88NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
  Strategy Concept notes
#89NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
Strategy Concept notes
#90MHT-CET → Maths → Determinants and Matrices → Determinants, Cofactors and the Adjoint Identities
Strategy Concept notes
Set · 1 question
Consider the following for the next two (02) items that follow: Let $\Delta(a,b,c,\alpha)=\begin{vmatrix}a&b&a\alpha+b\\b&c&b\alpha+c\\a\alpha+b&b\alpha+c&0\end{vmatrix}$ .
- #91NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
  Strategy Concept notes
#92MHT-CET → Maths → Determinants and Matrices → Determinants, Cofactors and the Adjoint Identities
Strategy Concept notes
#93NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
Strategy Concept notes
#94NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#95NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
Strategy Concept notes
#96MHT-CET → Maths → Determinants and Matrices → Determinants, Cofactors and the Adjoint Identities
Strategy Concept notes
#97MHT-CET → Maths → Determinants and Matrices → Determinants, Cofactors and the Adjoint Identities
Strategy Concept notes
#98NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
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#99NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
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