# Determinants

Exercise 6.1

Question 1

Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case:

(i) A =

Sol :

In a matrix, the minor is obtained for a particular element , by deleting that row and column where the element is present .

= (+1) (-1)

= -1

= -20

If A = then

Also ,

= -5

(ii) A =

Sol :

In a matrix, the minor is obtained for a particular element , by deleting that row and column where the element is present .

= (+1) (3)

= 3

= -4

If A = then

Also ,

= -11

(iii) A =

Sol :

In matrix , M_{ij} equals to the determinant of the sub-matrix obtained by leaving the i^{th} row and j^{th} column of *A* .

= – 2 – 10

= -12

= – 6 – 10

= -16

= – 6 + 2

= – 4

= (+1) (-12)

= -12

= 16

= – 4

Also , expanding the determinant along the first column .

= -12 + 6 + 46

= 40

(iv) A =

Sol :

Also , expanding the determinant along the first column .

(v) A =

Sol :

= 5

= 2 – 42

= – 40

= – 30

= 5

= 40

= – 30

Also , expanding the determinant along the first column .

= – 50

(vi) A =

Sol :

Also , expanding the determinant along the first column .

(vii) A =

Sol :

= – 1 – 8

= – 9

= 5 + 4

= 9

= – 10 + 1

= – 9

= +1 – 1

= 0

= – 9

= – 9

= – 9

= 0

Also , expanding the determinant along the first column .

= 0

Question 2

(i) A =

Sol :

(ii) A =

= 1

(iii) A =

Sol :

= cos 90°

= 0

(iv) A =

Sol :

Question 3

Evaluate :

(iv) A =

Sol :

Let

= 208 – 243 + 35

= 243 -243

= 0

= 0

Question 4

Show that

=1

Sol :

= sin 90°

= 1

**Alternate method**

and also

= 1

Question 5 (working)

Evaluate :

(iv) A = by two methods.

Sol :

**First method**

Question 6

Evaluate =

Question 7

Evaluate =

Question 8

If A = and B = , verify that =

Question 9

If A = ,then show that = 27

Question 10

Find the value of x, if

(i)

(ii)

(iii)

(iv) If = 10 , find the value of x .

Question 11

Find the integral value of x, if

Question 12

For what value of x the matrix A = is singular ?

Sol :