a b

Adeterminant D, i.e. D
=c d,l2x2 where a, b, c, d are certain numbers, they are calledthe
element of the determinant.
Value of determinant D = ad- bc.
An arrangement of mn numbers in the form of arectangular block of mmrows and n columns,
enclosed in rectangular brackets is called a matrix of the order m Xn.
Types of matrices: Zero matrix, Row matrix, Column matrix, Diagonal matrix, Null matrix,
Scalar matrix, Square matrix.
Associative law of matrix multiplication (AB) C= A(BC).
Inverse of matrix A denoted by A.
1
Inverse by adjoint method: A= IAI adj A.
Solve the following equations by matrix method.
1. 2x + 3y =9, -X+ y=-2
2. x + 3y =-2, 3x + 5y =4
3. X+ y= 1, 3y + 3z =5, 3z+3x = 4
4. X+y+Z= 1, 2x + y+ 27 =3,
3x +3y+ 4z =4
5. X+y+Z=6, 3x -y + 3z= 10,
5x + 5y- 4z =3
B.6.A. Usthéss Mathematics(Sem. ) 5.57 Matrices and Determinan
6. Find the values of following determinants.
1 2 3 2 3 4
() 4 6 5 67
(iü)
4 2 3 8 9 1
5 -2 0 X 1
(iii) -1 4 (iv)
-9 6
1 4 2 01 1

(v) 2 -1 4 (vi) 1 0 1
-3 -6 1 0
1

1 a b+c 1 11 1

(vii) 1 b c+a (viii) 1 1+x 1

1

1 c a+b 1 1 1+y
x +1 x+2 x+3 X 2 X+3

7. SoBve the equations: (i)| x +4 x +5 x+6 =0 (i) 3 5 = 0

x+7 x+8 x+1 7-x 12
2x + y 4
8. Find x and y if 5 x + 2y J-[]
+1 -2
9. If A =
A+ 2X= B.
2 3
B
-["
=
2
Find a matrix X satisfying the equation

10. Find 2A 3B.

3
I1. IA = -1 2
show that A'- 5A + 7L, = 0
3 -2
12.
ItAL4
= -2 satisfies the matrix equation A'- KA + 2I = 0, find k.

13. find |A - 21|

14. Solve for matrices X and Y if
3 -2 4 2
(i) 2X + 3Y
-[S 6
2 0
and 2X -3Y= -6
- | -1 2
and 2X + Y=
(ii) X-2Y 3
 -1 1
-2
15. IA=| 4 5
B =
3 6

Find AB and BA. Are they equal ?

4 -3
16. WA= , show that AA' =A'A=L,

17.

Show that BA' = (AB').

6
3 5
-[ 0 -1

Verify that A (BC) = (AB) C.

1 21 3

19. Find x, y, z if 0 3
-1 2 -2 4

20. Find the matrix X such that

6 2 4

4 X = 12 4 8
1 3 1 2J
X 4
21. If| is a singular matrix,find value of x.
2 8

Verify that |AB] = |A| |B|.

6 5
23. Show that A= 7 6
satisfies the equation A'-12A +I=0. Hence find A-!.

24. Find the adjoint of each of the following matrices.

12 2 2 0 -1 11 2 5
2 1 2 5 1 0 2 3 1
(iv)
2 2 1 I13J
1 -1
L 1 1
25. Find the adjoint of the matrix.
1 22 3
A = 0 5 0 and verify that I,. A (adj. A) = A| 1L (A
2 4 3
26. Findthe inverse of the following matrices :
2 -1 1

(ii) -1 2
6)
-1 2

f 2 0 -1 2 3 1

5 11 0 3 4 1
(iv)
01 3 3 7 2
3 2 4 6
Verify (AB)'= B-A-!
27. IfA =
75 -verity (AB)- ,
28. Solve the following equations.
(i) x+y+z= 6, 2x +y+ 22 = 10, 3x+ 3y + 4z = 21
(i) x +y+z= 1, 2x-y + 7z = 7 3x + y + 22 = 2
(iii) x +y+z=6, X-y +Z = 2, 2x + y -z= 1
(iv) 2x - y+z=1, X + 2y +32 = 8, 3x + y - 4z = 1
(v) 2x 4y + 3z = 1, x - 2y + 4z = 3, 3x-y + 5z = 2
(vi) -3x + 2y + z=3, 4x -y +3z = 11, x +y+4z = 8