ppr maths nbk

MATRICES

NOTES

Addition of Matrices

* ⎟⎟⎠

⎞⎜⎜⎝

⎛++++

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛sdrcqbpa

srqp

dcba

Subtraction of Matrices

* ⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛sdrcqbpa

srqp

dcba

Multiplication of a matrix by a number k

* k ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛kdkckbka

dcba

Multiplication of two matrices

1) ( ) ( )bqapqp

ba +=⎟⎟⎠

⎞⎜⎜⎝

⎛

2) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛bqbpaqap

qpba

3) ⎟⎟⎠

⎞⎜⎜⎝

⎛++

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛dqcpbqap

qp

dcba

4) ⎟⎟⎠

⎞⎜⎜⎝

⎛++++

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛dscqdrcpbsaqbrap

srqp

dcba

Inverse Matrix

If A = ⎟⎟⎠

⎞⎜⎜⎝

⎛dcba

, then inverse of A,

A-1 = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−− ac

bdbcad

1 ad – bc is known as determinant.

A-1 does not exist if the determinant is zero.

ppr maths nbk

EXERCISE 1

1) State the value of x if both of given matrices are equal

a) ⎟⎟⎠

⎞⎜⎜⎝

⎛− 453 x

, ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−4523

b) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

2324

x, ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−133

24x

2) Find the value of a and b for each of the following

a) ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛− b

a89

423

b) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−4197

2534

2363

ba

3) Find the value of p and q for each of the following

a) ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛qq

p28

35

23

3

b) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

+⎟⎟⎠

⎞⎜⎜⎝

⎛ −3

23

74

0143

22

4841 q

p

4) ( ) =−⎟⎟⎠

⎞⎜⎜⎝

⎛24

31

5) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛ −41

3451

=

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6) 2 ( ) ( ) ( )411234 =−+ yx , find the value of x + y

7) If ⎟⎟⎠

⎞⎜⎜⎝

⎛−

=+⎟⎟⎠

⎞⎜⎜⎝

⎛ −83

644032

M , then matrix M is

8) If the matrix ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−m263

does not have an inverse, find the value of m

9) If ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛1001

6732

N , find the matrix N

10) If A = ⎟⎟⎠

⎞⎜⎜⎝

⎛0231

, then A2 =

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ANSWER

1) a) x = -2

b) x = 21

−

2) a) a = 6 , b =41

b) a = 2 , b = 2

3) a) p = 6 , q = -6

b) p = 12 , q = 1

4) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

61224

5) ⎟⎟⎠

⎞⎜⎜⎝

⎛−)821

6) y = 1 , x = 3

7) ⎟⎟⎠

⎞⎜⎜⎝

⎛−123

92

8) m = 4

9) N = ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−

92

97

31

32

10) A2 = ⎟⎟⎠

⎞⎜⎜⎝

⎛6237

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Exercise 2

1. Given that the inverse matrix of 2 17 3⎛ ⎞⎜ ⎟⎝ ⎠

is m 37 2

n⎛ ⎞⎜ ⎟−⎝ ⎠

. Find the

values of m and n.

2. If A= 1 23 4

−⎛ ⎞⎜ ⎟−⎝ ⎠

, B= 1h

23 1

k⎛ ⎞⎜ ⎟−⎝ ⎠

and AB= 1 00 1⎛ ⎞⎜ ⎟⎝ ⎠

, find the values of h

and k.

3. If the matrix 23 6

z−⎛ ⎞⎜ ⎟⎝ ⎠

does not have an inverse, find the value of z.

4. If M 2 35 6⎛ ⎞⎜ ⎟⎝ ⎠

= 1 00 1⎛ ⎞⎜ ⎟⎝ ⎠

, find the matrix M.

5. Given that 9 6 1 6 1 012 1 9 0 1ba− −⎛ ⎞⎛ ⎞ ⎛ ⎞

=⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠, find the values of a and b.

6. (a) The inverse matrix of 3 24 5

−⎛ ⎞⎜ ⎟−⎝ ⎠

is k 5 23p

−⎛ ⎞⎜ ⎟⎝ ⎠

. Find the values of k

and p.

(b) Using the matrix method, solve the followind simultaneous equations. 3x – 2y = 12 4x – 5y = 23

7. (a) Find the inverse matrix of 3 51 4⎛ ⎞⎜ ⎟⎝ ⎠

(a) By using the matrix method, calculate the values of m and n

that satisfy the following simultaneous linear equations.

3m + 5n = 11 m + 4n = 13

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8. Given that matrix P= 1 23 1⎛ ⎞⎜ ⎟−⎝ ⎠

and PQ= 1 00 1⎛ ⎞⎜ ⎟⎝ ⎠

.

(a) Find the matrix Q.

(b) Hence, by using the matrix method, calculate the values of x and y that satisfy the following simultaneous equations.

x + 2y =8 3x – y =3

9. Given that the inverse of 3 25 4

−⎛ ⎞⎜ ⎟−⎝ ⎠

is m 45 3

n−⎛ ⎞⎜ ⎟−⎝ ⎠

.

(a) Find the values of m and n.

(b) Hence, by using the matrix method, calculate the values of x

and y that satisfy the following simultaneous equations.

3x – 2y = 8 5x – 4y = 13

10. Given that matrix P= 3 51 2⎛ ⎞⎜ ⎟−⎝ ⎠

and matrix Q=k 21 3

m−⎛ ⎞⎜ ⎟−⎝ ⎠

such that

PQ= 1 00 1⎛ ⎞⎜ ⎟⎝ ⎠

.

(a) Find the values of k and m,

(b) by using the matrix method, calculate the values of x and y that

satisfy the following simultaneous equations.

3x + 5y = 12 x – 2y = -7

ppr maths nbk

Answers (1) m = -1 (9) (a) m= - 1 n= 2

n = -1 2

(b) x= 3 y= 12

(2) h = 2 k = -4 (3) z = -4 (10) (a) k = - 1

11 , m = -5

(b) x = -1 , y = 3

(4) M = 6 315 23

−⎛ ⎞− ⎜ ⎟−⎝ ⎠

(5) a = 3

b = -2

(6) (a) k = - 17

, p = -4

(b) x = 2 , y = -3

(7) (a) 4 511 37

−⎛ ⎞⎜ ⎟−⎝ ⎠

(b) m = -3 , n = 4

(8) (a) 1 213 17− −⎛ ⎞−⎜ ⎟−⎝ ⎠

(b) x = 2 , y = 3

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DIAGNOSTIC TEST

1) Let matrix A = ⎟⎟⎠

⎞⎜⎜⎝

⎛963x

a) If the determinant for matrix A is zero, find the value of x

b) If x = 1,

i) find the inverse of matrix A

ii) using the matrix method , find the values of h and k that satisfy the

following simultaneous equation

h + 3k = -5

6h + 9k = 6

2) a) The inverse matrix of ⎟⎟⎠

⎞⎜⎜⎝

⎛4183

is ⎟⎟⎠

⎞⎜⎜⎝

⎛− 3141 t

k. Find the value of k and t.

b) Using matrices, calculate the values of x and y that satisfy the following

simultaneous linear equations

3x + 8y = 3

x + 4y = -1

3) Given that P = ⎟⎟⎠

⎞⎜⎜⎝

⎛h412

.

a) Calculate the value of h for which matrix P has no inverse matrix.

b) Given that h = -3, find the inverse matrix of P

c) Hence, calculate the values of x and y which satisfy the following matrix

equation.

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛− 11

334

12yx

ppr maths nbk

4. Given that matrix P= 6 44 2⎛ ⎞⎜ ⎟⎝ ⎠

and matrix Q= 1k

44 6

m −⎛ ⎞⎜ ⎟−⎝ ⎠

such that

PQ= 1 00 1⎛ ⎞⎜ ⎟⎝ ⎠

.

(c) Find the values of k and m,

(d) by using the matrix method, calculate the values of x and y that

satisfy the following matrix equation.

6 4 34 2 3

xy

⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎝ ⎠

5. Given the inverse matrix of 2 15 4

−⎛ ⎞⎜ ⎟−⎝ ⎠

is 45 2

kh

−⎛ ⎞⎜ ⎟−⎝ ⎠

.

(e) Find the values of h and k.

(b) By using the matrix method, calculate the values of x and y that satisfy the following simultaneous equations.

2x – y = 3 5x – 4y = 1

6. Given M is a 2X2 matrix where M 2 15 3

−⎛ ⎞⎜ ⎟−⎝ ⎠

= 1 00 1⎛ ⎞⎜ ⎟⎝ ⎠

.

(f) Find matrix M

(b) By using the matrix method, calculate the values of x and y that satisfy the following simultaneous equations.

2x – y = 7 5x – 3y = 19

ppr maths nbk

7. Given that matrix M= 2 14 3

⎛ ⎞⎜ ⎟−⎝ ⎠

and matrix Q= 1k

34 2

h⎛ ⎞⎜ ⎟⎝ ⎠

such that

MN= 1 00 1⎛ ⎞⎜ ⎟⎝ ⎠

.

(g) Find the values of k and h,

(h) by using the matrix method, calculate the values of x and y that

satisfy the following simultaneous linear equations.

2x + y =1 -4x + 3y = -17

8. Given that matrix A= 7 63 2

−⎛ ⎞⎜ ⎟−⎝ ⎠

, matrix B= 1k

23 7

v−⎛ ⎞⎜ ⎟−⎝ ⎠

, and AB=I, .

where I is the identity matrix

(i) Find the values of k and v.

(j) Hence, by using the matrix method, calculate the values of x and y that satisfy the following equation.

7 6 53 2 1

xy

−⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠

ppr maths nbk

ANSWER 1) a) x = 2

b) i) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−

91

32

311

ii) h = 7 , k = -4

2) a) k = 4 and t = -8

b) x = 5 and y = 23

−

3) a) h = 2

b) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−51

52

101

103

c) x = 2 and y = -1

4) (a) k = -4 , m = 2 (b) x = 3

2 , y = 3

2−

5) (a) h = 13− , k = 1

(b) x = 113

, y = 133

ppr maths nbk

6) (a) M = 3 15 2

−⎛ ⎞⎜ ⎟−⎝ ⎠

(b) x = 2 , y = -3

7) (a) k = 10 , h = -1 (b) x = 2 , y = -3 8) (a) k = 4 , v = 6

(b) x = -1 , y = -2