For a 2 × 2 matrix, like

2.jpg

Determinant is calculated like

Finding Determinant - Part 2

So,

     |A| = ad − bc

 

Let’s take an example

Finding Determinant - Part 3

 

For a 3 × 3 matrix, like

Finding Determinant - Part 4

Finding Determinant - Part 5

 

What about a 4 × 4 matrix?

Finding Determinant - Part 6

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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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.