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

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.

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