For a 2 × 2 matrix, like

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Determinant is calculated like

3.jpg

So,

     |A| = ad − bc

 

Let’s take an example

4.jpg

 

For a 3 × 3 matrix, like

5.jpg

6.jpg

 

What about a 4 × 4 matrix?

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  1. Chapter 4 Class 12 Determinants
  2. Concept wise

Transcript

A = [β– 8(𝒂&𝒃@𝒄&𝒅)] [β– 8(𝒂& 𝒃@𝒄& 𝒅)] Find determinant of A = [β– 8(3&2@1&4)] |A| = 3 Γ— 4 - 1 Γ— 2 = 12 βˆ’ 2 = 10 For a 3 Γ— 3 matrix, like A = [β– 8(𝒂&𝒃&𝒄@𝒅&𝒆&𝒇@π’ˆ&𝒉&π’Š)] |β– 8(𝒂&𝒃&𝒄@𝒅&𝒆&𝒇@π’ˆ&𝒉&π’Š)| = |β– 8( & @𝒆&𝒇@𝒉&π’Š)| βˆ’ |β– 8( & & @𝒅& &𝒇@π’ˆ& &π’Š)| + |β– 8( & @𝒅&𝒆@π’ˆ&𝒉)| |A| = a (ei βˆ’ hf) βˆ’ b (di βˆ’ gf) + c (dh βˆ’ eg) Note : There is a + βˆ’ pattern + βˆ’ + Let’s take an example Find determinant of B = [β– 8(9&2&3@5&βˆ’1&6@4&0&βˆ’2)] |B| = 9 Γ— |β– 8(βˆ’1&6@0&βˆ’2)| βˆ’2 Γ— |β– 8(5&6@4&βˆ’2)| + 1 Γ— |β– 8(5&βˆ’1@4&0)| = 9 ((βˆ’1) Γ— (βˆ’2) βˆ’ 0 Γ— 6) βˆ’ 2 (5 Γ— (βˆ’2) βˆ’4 Γ— 6) + 1 (5 Γ— 0 βˆ’ 4 Γ— (βˆ’1)) = 9 (2 βˆ’0) βˆ’ 2 (βˆ’10 βˆ’ 24) + 1 (0 + 4) = 9 Γ— 2 βˆ’ 2 Γ— (βˆ’34) + 1 Γ— 4 = 18 + 68 + 4 = 90 What about a 4 Γ— 4 matrix? For a 4 Γ— 4 matrix, like A = [β– 8(𝒂&𝒃&𝒄&𝒅@𝒆&𝒇&π’ˆ&𝒉@π’Š&𝒋&π’Œ&𝒍@π’Ž&𝒏&𝒐&𝒑)] Determinant is |β– 8( & & @𝒇&π’ˆ&𝒉@𝒋&π’Œ&𝒍@𝒏&𝒐&𝒑)| |β– 8( & & & @𝒆& &π’ˆ&𝒉@π’Š& &π’Œ&𝒍@π’Ž& &𝒐&𝒑)| |β– 8( & & & @𝒆& &π’ˆ&𝒉@π’Š& &π’Œ&𝒍@π’Ž& &𝒐&𝒑)| |β– 8( & & & @𝒆&𝒇& &𝒉@π’Š&𝒋& &𝒍@π’Ž&𝒏& &𝒑)| |β– 8( & & @𝒆&𝒇&π’ˆ@π’Š&𝒋&π’Œ@π’Ž&𝒏&𝒐)| Note : The + βˆ’ pattern is followed + βˆ’ + – Matrix Matrix is representation of number in row & column format Eg: A = [β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)] Matrix can be of any order [β– 8(3@5@6)]_(3 Γ— 1) [β– 8(3&2@1&4@5&3)]_(3 Γ— 2) [β– 8(3&2@1&4)]_(2 Γ— 2) Scalar multiplied to matrix If a number is multiplied to matrix, it is multiplied to each element of the matrix 2 [β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)] = [β– 8(2Γ—9&2Γ—2&2Γ—1@2Γ—5&2Γ—(βˆ’1)&2Γ—6@2Γ—4&2Γ—0&2Γ—(βˆ’2))] Determinant Determinant is number associated with a matrix Eg: |A| = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = 90 Determinant is only possible for a square matrix |β– 8(3&2@1&4@5&3)| Determinant not possible |β– 8(3&2@1&4)| Determinant possible Scalar multiplied to determinant If a number is multiplied to determinant, it is multiplied to either one row, or one column 2 |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = |β– 8(2Γ—9&2Γ—2&2Γ—1@5&βˆ’1&6@4&0&βˆ’2)| Or |β– 8(2Γ—9&2&1@2Γ—5&βˆ’1&6@2Γ—4&0&βˆ’2)|

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.