Finding determinant of a 2x2 matrix

Chapter 4 Class 12 Determinants
Concept wise

For a 2 Γ 2 matrix, like

Determinant is calculated like

So,

Β  Β  Β |A| = ad β bc

Β

Letβs take an example

Β

Β

## What about a 4 Γ 4 matrix?

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### Transcript

A = [β 8(π&π@π&π)] [β 8(π& π@π& π)] Find determinant of A = [β 8(3&[email protected]&4)] |A| = 3 Γ 4 - 1 Γ 2 = 12 β 2 = 10 For a 3 Γ 3 matrix, like A = [β 8(π&π&π@π&π&π@π&π&π)] |β 8(π&π&π@π&π&π@π&π&π)| = |β 8( & @π&π@π&π)| β |β 8( & & @π& &π@π& &π)| + |β 8( & @π&π@π&π)| |A| = a (ei β hf) β b (di β gf) + c (dh β eg) Note : There is a + β pattern + β + Letβs take an example Find determinant of B = [β 8(9&2&[email protected]&β1&[email protected]&0&β2)] |B| = 9 Γ |β 8(β1&[email protected]&β2)| β2 Γ |β 8(5&[email protected]&β2)| + 1 Γ |β 8(5&β[email protected]&0)| = 9 ((β1) Γ (β2) β 0 Γ 6) β 2 (5 Γ (β2) β4 Γ 6) + 1 (5 Γ 0 β 4 Γ (β1)) = 9 (2 β0) β 2 (β10 β 24) + 1 (0 + 4) = 9 Γ 2 β 2 Γ (β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(π&π&π&π@π&π&π&π@π&π&π&π@π&π&π&π)] Determinant is |β 8( & & @π&π&π@π&π&π@π&π&π)| |β 8( & & & @π& &π&π@π& &π&π@π& &π&π)| |β 8( & & & @π& &π&π@π& &π&π@π& &π&π)| |β 8( & & & @π&π& &π@π&π& &π@π&π& &π)| |β 8( & & @π&π&π@π&π&π@π&π&π)| Note : The + β pattern is followed + β + β Matrix Matrix is representation of number in row & column format Eg: A = [β 8(9&2&[email protected]&β1&[email protected]&0&β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]&β1&[email protected]&0&β2)] = [β 8(2Γ9&2Γ2&2Γ[email protected]Γ5&2Γ(β1)&2Γ[email protected]Γ4&2Γ0&2Γ(β2))] Determinant Determinant is number associated with a matrix Eg: |A| = |β 8(9&2&[email protected]&β1&[email protected]&0&β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]&β1&[email protected]&0&β2)| = |β 8(2Γ9&2Γ2&2Γ[email protected]&β1&[email protected]&0&β2)| Or |β 8(2Γ9&2&[email protected]Γ5&β1&[email protected]Γ4&0&β2)|