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**F. DETERMINANT**

**(i) Submatrix :** Let A be a given matrix. The matrix obtained by deleting some rows or columns of A is called as submatrix of A.

e.g.

are all submatrices of A.

**(ii) Determinant of A Square Matrix :**

Let A [a]_{1 x 1} be a 1 x 1 matrix. Determinant A is defined as |A| = a eg. A = [â€“3]_{1 x 1} |A| = â€“3

,then |A| is defined as ad â€“bc.

**(iii)Minors & Cofactors **: Let Î” be a determinant. Then minor of element a_{ij}, denoted by M_{ij} is defined as the determinant of the submatrix obtained by deleting i^{th} row & j^{th} column of Î”.

Cofactor of element a_{ij}, denoted by C_{ij} is defined as C_{ij} = (â€“1)^{i + j} M_{ij}.

**(iv)Determinant **: Let A = [a_{ij}]_{n} be a square matrix (n > 1). Determinant of A is defined as the sum of products of elements of any one row (or one column) with corresponding cofactors.

|A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} (using first row)

|A| = a_{12 }C_{12} + a_{22}C_{22} + a_{32}C_{32} (using second column)

**G. PROPERTIES OF DETERMINANTS**

**P- 1 :** The value of a determinant remains unaltered , if the rows & columns are inter changed . e.g. If

are transpose of each other.

If D' = - D then it is SKEW SYMMETRIC determinant but D' = D â‡’ 2 D = 0 â‡’ D = 0 â‡’ Skew symmetric determinant of third order has the value zero .

**P-2** : If any two rows (or columns) of a determinant be interchanged , the value of determinant is changed in sign only . e.g.

**P-3** : If a determinant has any two rows (or columns) identical , then its value is zero.

**P-4** : If all the elements of any row (or column) be multiplied by the same number then the determinant is multiplied by that number.

e.g. If

**P-5** : If each element of any row (or column) can be expressed as a sum of two terms then the determinant can be expressed as the sum of two determinants.

**P-6** : The value of a determinant is not altered by adding to the elements of any row (or column) the same multiples of the corresponding elements of any other row (or column).**Note that** while applying this property atleast one row (or column) must remain unchanged.

**P- 7** : If by putting x = a the value of a determinant vanishes then (x-a) is a factor of the determinant

**Ex.17 Find the value of the determinant**

**Sol.**

**Ex.18 ****A is a n Ã— n matrix (n > 2) [a _{ij}] where **

**Sol.**

â‡’ value of determinant is zero.

**H. MULTIPLICATION OF TWO DETERMINANTS**

(i)

Similarly two determinants of order three are multiplied.

(ii) where A_{i} , B_{i} , C_{i} are cofactors

PROOF : Consider

Note : a_{1}A_{2} + b_{1}B_{2} + c_{1}C_{2 }= 0 etc.

therefore

**Ex.19 Prove that **

= (a â€“ b) (a â€“ c) (a â€“ d) (b â€“ c) (b â€“ d) (c â€“ d).

**Sol.**

Applying R_{2} â†’ R_{2}- R_{1}, R_{3} â†’ R_{3 }- R_{1}, we get

=(a â€“ b) (c â€“ d) (a â€“ c) (b â€“ d)

= (a â€“ b) (c â€“ d) (a â€“ c) (b â€“ d) [(a + c) (b + d) â€“ (a + b) (c + d)]

= (a â€“ b) (c â€“ d) (a â€“ c) (b â€“ d) (ab +cd â€“ ac â€“ bd) = (a â€“ b) (a â€“ c) (a â€“ d) (b â€“ c) (b â€“ d) (c â€“ d).

**Alternatively : **

and using c_{3} â†’ c_{3} + 2 a b c d . c_{3}

**Ex.20 Show that**

(where r^{2} = x^{2} + y^{2 }+ z^{2} & u^{2} = x y + y z + z x)

**Sol. **Consider the determinant , We see that the L.H.S. determinant has its constituents

which are the co-factor of Î”. Hence L.H.S. determinant

**Ex.21 Without expanding, as for as possible, prove that**

= (x â€“ y) (y â€“ z) (z â€“ x) (x + y + z)

**Sol.**

for x = y, D = 0 (since C_{1} and C_{2} are identical)

Hence (x â€“ y) is a factor of D (y â€“ z) and (z â€“ x) are factors of D. But D is a homogeneous expression of the 4th degree is x, y, z.

âˆ´ There must be one more factor of the 1st degree in x, y, z say k (x + y + z) where k is a constant.

Let D = k (x â€“ y) (y â€“ z) (z â€“ x) (x + y + z),Putting x = 0, y = 1, z = 2

= k (0 â€“ 1) (1 â€“ 2) (2 â€“ 0) (0 + 1 + 2)

â‡’ L(8 â€“ 2) = k(â€“1) (â€“1) (2) (3) âˆ´ k = 1 âˆ´ D = (x - y) (y - z) (z - x) (x + y + z)

**Ex.22 Prove that**

**Sol.**

Given that Î” =

**Ex.23 Express** **as product of two determinants.**

**Sol. **The given determinant is

with the help of rowâ€“byâ€“row multiplication rule.

**Ex.24 **

**Express the determinant D as a product of two determinants. Hence or otherwise show that D = 0.**

**Sol.**

as can be seen by applying rowâ€“byâ€“row multiplication rule., Hence D = 0.

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