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Finding Determinant

Last updated at March 16, 2023 by

For a 2 Γ— 2 matrix, like

2.jpg

Determinant is calculated like

Finding Determinant - Part 2

So,

Β  Β  Β |A| = ad βˆ’ bc

Β 

Let’s take an example

Finding Determinant - Part 3

Β 

For a 3 Γ— 3 matrix, like

Finding Determinant - Part 4

Finding Determinant - Part 5

Β 

What about a 4 Γ— 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&[email protected]&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&[email protected]&βˆ’1&[email protected]&0&βˆ’2)] |B| = 9 Γ— |β– 8(βˆ’1&[email protected]&βˆ’2)| βˆ’2 Γ— |β– 8(5&[email protected]&βˆ’2)| + 1 Γ— |β– 8(5&βˆ’[email protected]&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&[email protected]&βˆ’1&[email protected]&0&βˆ’2)] Matrix can be of any order [β– 8([email protected]@6)]_(3 Γ— 1) [β– 8(3&[email protected]&[email protected]&3)]_(3 Γ— 2) [β– 8(3&[email protected]&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&[email protected]&βˆ’1&[email protected]&0&βˆ’2)] = [β– 8(2Γ—9&2Γ—2&2Γ—[email protected]Γ—5&2Γ—(βˆ’1)&2Γ—[email protected]Γ—4&2Γ—0&2Γ—(βˆ’2))] Determinant Determinant is number associated with a matrix Eg: |A| = |β– 8(9&2&[email protected]&βˆ’1&[email protected]&0&βˆ’2)| = 90 Determinant is only possible for a square matrix |β– 8(3&[email protected]&[email protected]&3)| Determinant not possible |β– 8(3&[email protected]&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&[email protected]&βˆ’1&[email protected]&0&βˆ’2)| = |β– 8(2Γ—9&2Γ—2&2Γ—[email protected]&βˆ’1&[email protected]&0&βˆ’2)| Or |β– 8(2Γ—9&2&[email protected]Γ—5&βˆ’1&[email protected]Γ—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.