Mathematics Revision Notes of Determinants for NDA

Determinants Introduction

Every square matrix `A= [a_(ij)]` of order `n` can be associated to an expression or a number which is called determinant. It is denoted by `|A| ` or `det A` or `Delta` If `A = [(a,b), (c,d)]` Then determinant of `A` is written as

`|A|=|(a,b),(c,d)| = det(A)`

`"Note "` `=>` For matrix `A, |A|` is read as determinant of `A` not modulus of `A` .

`=>` Only square matrices have determinant.



Determinants of a Square Matrix of Order `1`

Let `A= |a_(11) |` be a `1 xx 1` matrix , then the determinants of `A` is the number `a_(11)` itself i.e., `|A| =a_(11)`

Determinant of a Square Matrix of Order `2`

Let `A= [(a_(11) , a_(12)), (a_(21), a_(22))]` be a matrix of order `2 xx 2`, then

`det (A) ` or `|A| ` or `Delta =|(a_(11) , a_(12)) ,(a_(21), a_(22))|=a_(11)a_(22)- a_(21) a_(12)`

Determinant of Matrix of Order 3

Let `|A| =[ (a_(11) , a_(12), a_(13)), (a_(21) , a_(22), a_(23)), (a_(31), a_(32), a_(33))]` be a square matrix of order `3`

Then, `|A| = |(a_(11), a_(12), a_(13)), (a_(21), a_(22), a_(23)), (a_(31), a_(32), a_(33))|=> |A| =a_(11) |(a_(22), a_(23)), (a_(32), a_(33))|- a_(12) |(a_(21), a_(23)), (a_(31), a_(33))|+ a_(13) |(a_(21), a_(22)), (a_(31), a_(32))|`

`= a_(11)(a_(22) a_33 - a_23 a_32)- a_12(a_21 a_33 - a_23 a_31)+ a_13 (a_21 a_32 - a_22 a_31)`

Minors of a Determinant

Minor of an element is the determinant obtained by deleting the row and column in which that element lies. It is denoted by `M_(ij)`

e.g. Given, a `3 xx 3` determinant

`|(a_31, a_32, a_33),(a_41, a_42, a_43), (a_51, a_52, a_53)|`

Minor of `a_31 = |(a_42, a_43), (a_52, a_43)|`

and minor of `a_32` is `M_32= |(a_41, a_43),(a_51, a_53)|`

Similarly, it can be calculated for other elements of a determinant.

Cofactors of a Determinant

Let `A = { a_(ij)}` be an `n` order square matrix. Then, the cofactor `C_(ij)` of `a_(ij)` for `A` will be `(- 1)^(i+j) ` times of `M_(ij)` (minor of an element of a determinant). It means `C_(ij)= (- 1)^(i+j) M_(ij)`

`C_(ij) = { tt((-M_(ij) , when , i+j , is, odd),(M_(ij) , when , i+j , is , even))`

e.g. Cofactor of `a_31` in `3 xx 3` determinant `|(a_31 , a_32, a_33),(a_41, a_42, a_43),(a_51, a_52, a_53)|` will be `C_31`

i.e., `C_31 =(-1)^(3+1) |(a_42 , a_43),(a_52 , a_52)|`

Properties of Determinants

1. If each element of a row (column) is a zero, then `Delta =0`.

2. If two rows (columns) are proportional, then `Delta = 0`.

3. If any two rows (columns) are interchanged odd times, then `Delta` becomes `-Delta`

4. If the rows and columns are interchanged, then `Delta` is unchanged i.e. `|A^T|= |A |`.

5. If each element of a row (column) of a determinant is multiplied by a constant `k`, then the value of the new determinant is `k` times the value of the original determinant

i.e., `|(ka , kb , kc),(p, q, r),(u, v, w)|= k |(a, b, c),(p, q, r), (u, v , w)|`


6. `|(a_1+a_2 , b , c),(p_1+p_2 , q, r), (u_1+u_2 , v , w)|=|(a_1 , b, c), (p, q , r),(u_1 , v , w)| + | (a_2 , b , c), (p_2 , q , r), (u_2 , v , w)|`


7. If a scalar multiple of any row (column) is added to another row (column), then `Delta` is unchanged

i.e., `|(a, b, c) , (p, q , r), (u, v, w)| = | (a, b, c), (p+ka, q+kb , r+kc), (u, v, w)|`


8. Let `Delta(x)` be a `3rd` order determinant having polynomials as its elements.

(i) If `Delta (a)` has two rows (columns) proportional, then `(x- a)` is a factor of `Delta (x)`.

(ii) If `Delta (a)` has three rows (columns) proportional, then `(x- a )^2` is a factor of `Delta (x)`.

9. Product of two determinants

i.e., `|AB|= |A||B| =|BA| =|AB^T| =|A^T B| =|A^T B^T|`

10. If `Delta( x) =|(f_1 (x), f_2(x) , f_3(x)), (g_1(x) , g_2(x), g_3(x)), (a, b, c)|` then

(i) `sum_(x=1)^n Delta (x) = |(sum_(x=1)^n f_1(x) , sum_(x=1)^n f_2(x) , sum_(x=1)^n f_3(x)), (sum_(x=1)^n g_1(x) , sum_(x=1)^n g_2(x) , sum_(x=1)^n g_3(x)), (a, b, c)|`

(ii) `prod_(x=1)^n Delta (x) = |(prod_(x=1)^n f_1(x) , prod_(x=1)^n f_2(x) , prod_(x=1)^n f_3(x)), (prod_(x=1)^n g_1(x) , prod_(x=1)^n g_2(x) , prod_(x=1)^n g_3(x)),(a, b, c)|`


11. `det(kA)=k^n det (A) `, if `A` is of order `n xx n`.

12. `det (A^n) =(det A)^n `, if `n in I^+`

13. `|A^T| =|A|` where `A^T` is a transpose of a matrix.

`"Symmetric Determinant:"`

`|(a, h, g),(h,b,f),(g,f,c)|= abc + 2 fgh - af^2-bg^2- ch^2`

`"Skew-symmetric Determinant"`

`|(0, h, -g), (-h, 0, f), (g, -f, 0)|=0`

In skew-symmetric matrix of odd order, the value of determinant is zero.

Product of Determinants

Generally, the product of determinants can be calculated by row column multiplication rule and it is same as the rule of multiiplication of two matrices.

e.g., Let `Delta_1= |(a_1, b_1 , c_1),(a_2, b_2, c_2), (a_3 , b_3, c_3)|` and `Delta_2= |(alpha_1 , beta_1 , gamma_1), (alpha_2 , beta_2 , gamma_2),(alpha_3, beta_3, gamma_3)|`

be two determinants , then

`Delta_1 Delta_2= |(a_1alpha_1+b_1alpha_2+c_1 alpha_3 ,a_1 beta_1+b_1 beta_2+c_1 beta_3, a_1 gamma_1+b_1 gamma_2+c_1 gamma_3),(a_2 alpha_1+b_2alpha_2+c_2 alpha_3 ,a_2 beta_1+b_2 beta_2+c_2 beta_3, a_2 gamma_1+b_2 gamma_2+c_2 gamma_3), (a_3alpha_1+b_3alpha_2+c_3 alpha_3 ,a_3 beta_1+b_3 beta_2+c_3 beta_3, a_3 gamma_1+b_3 gamma_2+c_3 gamma_3)|`

Cyclic Determinant

A circulant matrix is one in which each row vector is rotated one element to the right relative to the preceding
row vector.

1. `|(1, x, x^2), (1, y, y^2), (1, z, z^2)| =(x-y) (y-z)(z-x )`

2. `| (1, x, x^3), (1, y, y^3), (1, z, z^3)| = (x - y) (y - z) (z - x ) (x + y + z)`

3. `| (1, x^2, x^3), (1, y^2, y^3), (1, z^2, z^3)| = (x - y) (y - z) (z - x ) (xy + yz + zx)`

4. `| (a, b, c), (b, c , a), (c , a , b)| = - (a+b+c) (a^2 + b^2 + c^2 - ab - bc - ca)`

` = - (a^3 + b^3 + c^3 - 3abc)`

5. `| (a, bc, abc), (b, ca , abc), (c , ab , abc)| = | (a, a^2, a^3), (b, b^2 , b^3), (c , c^2 , b^3)| =abc(a-b )(b -c )(c -a)`

6. ` | ( cos(A - P) , cos(A - Q) ,cos (A - R) ),(cos(B- P) , cos(B- Q) , cos(B- R)) , (cos(C- P), cos(C- Q) , cos(C- R))| = 0`

7. ` | ( 1 , cos(beta - alpha ) , cos (gamma -alpha )) ,
(cos( alpha - beta ) , 1 , cos(gamma - beta ) ) ,
( cos ( alpha - gamma ) , cos ( beta - gamma) , 1 ) | = 0`

8. ` | ( (a_1 - b_1)^2 , (a_1 - b_2)^2 , (a_1 - b_3)^2 ),( (a_2 - b_1)^2 , (a_2 - b_2)^2 , (a_2 - b_3)^2),( (a_3 - b_1)^2 , (a_3 - b_2)^2 , (a_3 - b_3)^2)| = 0`

` = 2 (a_1 - a_2) (a_2 - a_3) (a_3 - a_1) ( b_1 - b_2) (b_2 - b_3) (b_3 - b_1)`

Area of Triangles using Determinants

If three vertices of a triangle are `(x_1 , y_1 ), (x_2 , y_2)` and `(x_3, y_3)`,

then the area of the triangle can be calculated as

` 1/2 | (x_1, y_1 , 1 ),( x_2, y_2 , 1 ), (x_3 , y_3 , 1 ) | = 1/2 | { x_1 (y_2 - y_3 ) + x_2 ( y_3 - y_1) + x_3 (y_1 - y_2 )}|`

`"NOTE"` Condition for three points to be colliner, is given by

` | (x_1, y_1 , 1 ),( x_2, y_2 , 1 ), (x_3 , y_3 , 1 ) | = 0 `

Solution of System of Linear Equations

For the solution of system of non-homogeneous equation in two or three variables.

Let `a_1 x + b_1 y + c_1 z = d_1`

`a_2 x + b_2 y + c_2 z = d_2` and `a_3 x + b_3 y + c_3 z = d_3`

be a system of linear equations.
Then, the set of values of variables x, y and z which simultaneously satisfy these three equations, is called a solution.


`"Consistent"`

If the system of equations has a unique solution or infinite many solutions, then the system of equations is known as consistent.

`"Inconsistent"`
If the system of equations has no solution, then the system of equations is known as inconsistent.

Cramer's Rule

A system of simultaneous linear equations can be solved by Cramer's rule,. named after the Swiss Mathematician Gabriel Cramer.

(i) In Two Variables Let we have two equations
`a_1 x + b_1 y = c_1` and `a_2 x + b_2 y = c_2`
Then, its solution will be

` x = D_1 /D ` and ` y = D_2 /D `

where, ` D = | (a_1 , b_1),(a_2 , b_2) | , D_1 = | (c_1 , b_1),(c_2 , b_2) | `

and ` D_2 = | (a_1 , c_1),(a_2 , c_2) | ` , provided that `D != 0`.

(ii) In Three Variables Let we have three equations

`a_1 x + b_1 y + c_1 z = d_1`

`a_2 x + b_2 y + c_2 z = d_2`

`a_3 x + b_3y + c_3 z = d_3`

Then, .its solution will be

` x = D_1 /D ` and ` y = D_2 /D `

and ` z = D_3 /3 `

where, ` D = | ( a_1 , b_1 , c_1),( a_2 , b_2 , c_2),( a_3 , b_3 , c_3) | , D_1 = | ( d_1 , b_1 , c_1),( d_2 , b_2 , c_2),( d_3 , b_3 , c_3) | `

`D_2 = | ( a_1 , d_1 , c_1),( a_2 , d_2 , c_2),( a_3 , d_3 , c_3) | `

`D_3 = | ( a_1 , b_1 , d_1),( a_2 , b_2 , d_2),( a_3 , b_3 , d_3) | `

provided that `D != 0`.

Conditions for Consistency

`"Non - homogeneous System"`
The following cases arise while solving linear equations in two or three variables :
(i) If `D != 0`, then system will be consistent for a unique solution.
(ii) If `D = 0` and at least one of `D_1, D_2` and `D_3` is non - zero, then given system is inconsistent i.e. no solution.
(iii) If `D = 0` and `D_1 = D_2 = D_3 = 0`, then given system is
consistent with infinite many solutions or is inconsistent.

`"Homogeneous System"`
The following cases arise while solving linear equations in two or three variables :
(i) When `D != 0`, then system has trivial solution.
i.e. `x = y = z = 0`
(ii) When `D = 0`, then system has non-trivial solution i.e. infinitely many solutions .

 
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