1. ppr maths nbk
MATRICES
NOTES
Addition of Matrices
⎛a b ⎞ ⎛ p q⎞ ⎛a + p b + q⎞
*⎜
⎜c d ⎟ + ⎜ r s ⎟ = ⎜ c + r d + s⎟
⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Subtraction of Matrices
⎛a b ⎞ ⎛ p q⎞ ⎛a − p b − q⎞
*⎜
⎜c d ⎟ − ⎜ r s ⎟ = ⎜ c − r d − s⎟
⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Multiplication of a matrix by a number k
⎛ a b ⎞ ⎛ ka kb ⎞
* k⎜
⎜ c d ⎟ = ⎜ kc kd ⎟
⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Multiplication of two matrices
⎛ p⎞
1) (a b )⎜ ⎟ = (ap + bq )
⎜q⎟
⎝ ⎠
⎛a⎞ ⎛ ap aq ⎞
2) ⎜ ⎟( p q ) = ⎜
⎜b⎟ ⎜ bp bq ⎟
⎟
⎝ ⎠ ⎝ ⎠
⎛ a b ⎞⎛ p ⎞ ⎛ ap + bq ⎞
3) ⎜
⎜ c d ⎟⎜ q ⎟ = ⎜ cp + dq ⎟
⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ a b ⎞⎛ p q ⎞ ⎛ ap + br aq + bs ⎞
4) ⎜
⎜ c d ⎟⎜ r s ⎟ = ⎜ cp + dr
⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ cq + ds ⎟
⎠
Inverse Matrix
⎛a b ⎞
If A = ⎜
⎜ c d ⎟ , then inverse of A,
⎟
⎝ ⎠
1 ⎛ d − b⎞
A-1 = ⎜ ⎟ ad – bc is known as determinant.
ad − bc ⎜ − c a ⎟
⎝ ⎠
A-1 does not exist if the determinant is zero.

2. ppr maths nbk
EXERCISE 1
1) State the value of x if both of given matrices are equal
⎛ 3 x ⎞ ⎛ 3 − 2⎞
a) ⎜
⎜ − 5 4⎟ , ⎜ − 5 4 ⎟
⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛4 − 2 ⎞ ⎛4 − 2 ⎞
b) ⎜
⎜ 3 x − 2⎟ ,
⎟ ⎜
⎜ 3 3 x − 1⎟
⎟
⎝ ⎠ ⎝ ⎠
2) Find the value of a and b for each of the following
⎛ 3 ⎞ ⎛a⎞ ⎛ 9 ⎞
a) ⎜ ⎟ + ⎜ ⎟ = ⎜ ⎟
⎜ − 2 ⎟ ⎜ 4 ⎟ ⎜ 8b ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ − 3 6a ⎞ ⎛ 4 3 ⎞ ⎛ − 7 9 ⎞
b) ⎜
⎜ 3b 2 ⎟ − ⎜ 5 − 2 ⎟ = ⎜ 1 4 ⎟
⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
3) Find the value of p and q for each of the following
⎛ p⎞ ⎛ 5 ⎞ ⎛ 8 ⎞
a) 3⎜ ⎟ − 2⎜ ⎟ = ⎜ ⎟
⎜ 3q ⎟ ⎜ − 3 ⎟ ⎜ 2q ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
1 ⎛ 8 − 4 ⎞ ⎛ − 3 4q ⎞ ⎛ − 4
⎜
7⎞
⎟
b) ⎜
⎜ 2 p ⎟ + 2⎜ − 1 0 ⎟ = ⎜ − 3
⎟ ⎜ ⎟
4⎝ ⎠ ⎝ ⎠ ⎝ 2 3⎟
⎠
⎛1⎞
4) ⎜ ⎟(4 − 2 ) =
⎜ 3⎟
⎝ ⎠
⎛ 1 − 5 ⎞⎛ − 1⎞
5) ⎜
⎜ 4 3 ⎟⎜ 4 ⎟ =
⎟⎜ ⎟
⎝ ⎠⎝ ⎠

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6) 2 (4 x ) + y (3 − 2 ) = (11 4 ) , find the value of x + y
⎛ 2 − 3⎞ ⎛4 6 ⎞
7) If ⎜
⎜ 0 4 ⎟ + M = ⎜ 3 − 8 ⎟ , then matrix M is
⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ − 3 − 6⎞
8) If the matrix ⎜
⎜ 2 ⎟ does not have an inverse, find the value of m
⎝ m⎟ ⎠
⎛ 2 3⎞ ⎛1 0⎞
9) If ⎜
⎜7 6⎟⎟N = ⎜
⎜ 0 1 ⎟ , find the matrix N
⎟
⎝ ⎠ ⎝ ⎠
⎛ 1 3⎞
10) If A = ⎜
⎜ 2 0 ⎟ , then A =
⎟
2
⎝ ⎠

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ANSWER
1) a) x = -2
1
b) x = −
2
1
2) a) a = 6 , b =
4
b) a = 2 , b = 2
3) a) p = 6 , q = -6
b) p = 12 , q = 1
⎛ 4 − 2⎞
4) ⎜
⎜12 − 6 ⎟
⎟
⎝ ⎠
⎛ − 21⎞
5) ⎜
⎜ 8) ⎟⎟
⎝ ⎠
6) y = 1 , x = 3
⎛2 9 ⎞
7) ⎜
⎜ 3 − 12 ⎟
⎟
⎝ ⎠
8) m = 4
⎛−2 1 ⎞
⎜ ⎟
9) N = ⎜ 3 3 ⎟
⎜ 7 −2⎟
⎜ ⎟
⎝ 9 9 ⎠
⎛ 7 3⎞
10) A2 = ⎜
⎜ 2 6⎟
⎟
⎝ ⎠

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Exercise 2
⎛ 2 1⎞ ⎛ 3 n⎞
1. Given that the inverse matrix of ⎜ ⎟ is m ⎜ ⎟ . Find the
⎝ 7 3⎠ ⎝ −7 2 ⎠
values of m and n.
⎛ 1 −2 ⎞ 1 ⎛ k 2⎞ ⎛1 0⎞
2. If A= ⎜ ⎟ , B= ⎜ ⎟ and AB= ⎜ ⎟ , find the values of h
⎝ 3 −4 ⎠ h ⎝ −3 1 ⎠ ⎝0 1⎠
and k.
⎛ −2 z ⎞
3. If the matrix ⎜ ⎟ does not have an inverse, find the value of z.
⎝ 3 6⎠
⎛ 2 3⎞ ⎛1 0⎞
4. If M ⎜ ⎟ =⎜ ⎟ , find the matrix M.
⎝ 5 6⎠ ⎝ 0 1⎠
1 ⎛ −9 −6 ⎞ ⎛ 1 6 ⎞ ⎛ 1 0 ⎞
5. Given that ⎜ ⎟⎜ ⎟=⎜ ⎟ , find the values of a and b.
a ⎝ 2 1 ⎠ ⎝ b −9 ⎠ ⎝ 0 1 ⎠
⎛ 3 −2 ⎞ ⎛ −5 2 ⎞
6. (a) The inverse matrix of ⎜ ⎟ is k ⎜ ⎟ . Find the values of k
⎝ 4 −5 ⎠ ⎝ p 3⎠
and p.
(b) Using the matrix method, solve the followind simultaneous
equations.
3x – 2y = 12
4x – 5y = 23
⎛3 5⎞
7. (a) Find the inverse matrix of ⎜ ⎟
⎝1 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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⎛1 2 ⎞ ⎛1 0⎞
8. Given that matrix P= ⎜ ⎟ and PQ= ⎜ ⎟.
⎝ 3 −1⎠ ⎝0 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
⎛ 3 −2 ⎞ ⎛ −4 n ⎞
9. Given that the inverse of ⎜ ⎟ is m ⎜ ⎟.
⎝ 5 −4 ⎠ ⎝ −5 3 ⎠
(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
⎛3 5 ⎞ ⎛ −2 m ⎞
10. Given that matrix P= ⎜ ⎟ and matrix Q=k ⎜ ⎟ such that
⎝ 1 −2 ⎠ ⎝ −1 3 ⎠
⎛1 0⎞
PQ= ⎜ ⎟.
⎝0 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

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Answers
(1) m = -1 (9) (a) m= - 1 n= 2
n = -1 2
1
(b) x= 3 y=
2
(2) h = 2
k = -4
1
(3) z = -4 (10) (a) k = - , m = -5
11
(b) x = -1 , y=3
1⎛ 6 −3 ⎞
(4) M= − ⎜ ⎟
3 ⎝ −5 2⎠
(5) a=3
b = -2
1
(6) (a) k = - , p = -4
7
(b) x = 2 , y = -3
1 ⎛ 4 −5 ⎞
(7) (a) ⎜ ⎟
7 ⎝ −1 3 ⎠
(b) m = -3 , n = 4
−1 ⎛ −1 −2 ⎞
(8) (a) ⎜ ⎟
7 ⎝ −3 1 ⎠
(b) x = 2 , y = 3

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DIAGNOSTIC TEST
⎛ x 3⎞
1) Let matrix A = ⎜
⎜ 6 9⎟
⎟
⎝ ⎠
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
⎛3 8⎞ 1⎛ 4 t⎞
2) a) The inverse matrix of ⎜ ⎟ is ⎜
⎜1 4⎟ ⎟ . Find the value of k and t.
⎝ ⎠ k ⎜ − 1 3⎟
⎝ ⎠
b) Using matrices, calculate the values of x and y that satisfy the following
simultaneous linear equations
3x + 8y = 3
x + 4y = -1
⎛2 1⎞
3) Given that P = ⎜
⎜ 4 h⎟ .
⎟
⎝ ⎠
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.
⎛ 2 1 ⎞⎛ x ⎞ ⎛ 3 ⎞
⎜
⎜ 4 − 3 ⎟⎜ y ⎟ = ⎜11⎟
⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠

9. ppr maths nbk
⎛ 6 4⎞ 1 ⎛ m −4 ⎞
4. Given that matrix P= ⎜ ⎟ and matrix Q= ⎜ ⎟ such that
⎝ 4 2⎠ k ⎝ −4 6 ⎠
⎛1 0⎞
PQ= ⎜ ⎟.
⎝0 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 ⎞⎛ x ⎞ ⎛ 3 ⎞
⎜ ⎟⎜ ⎟ = ⎜ ⎟
⎝ 4 2 ⎠⎝ y ⎠ ⎝ 3 ⎠
⎛ 2 −1 ⎞ ⎛ −4 k ⎞
5. Given the inverse matrix of ⎜ ⎟ is h ⎜ ⎟.
⎝ 5 −4 ⎠ ⎝ −5 2 ⎠
(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
⎛ 2 −1 ⎞ ⎛ 1 0 ⎞
6. Given M is a 2X2 matrix where M ⎜ ⎟ =⎜ ⎟.
⎝ 5 −3 ⎠ ⎝ 0 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

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⎛ 2 1⎞ 1 ⎛3 h⎞
7. Given that matrix M= ⎜ ⎟ and matrix Q= ⎜ ⎟ such that
⎝ −4 3 ⎠ k ⎝4 2⎠
⎛1 0⎞
MN= ⎜ ⎟.
⎝0 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
⎛ 7 −6 ⎞ 1 ⎛ −2 v ⎞
8. Given that matrix A= ⎜ ⎟ , matrix B= ⎜ ⎟ , and AB=I, .
⎝ 3 −2 ⎠ k ⎝ −3 7 ⎠
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 ⎞ ⎛ x ⎞ ⎛ 5 ⎞
⎜ ⎟⎜ ⎟ = ⎜ ⎟
⎝ 3 −2 ⎠ ⎝ y ⎠ ⎝ 1 ⎠

11. ppr maths nbk
ANSWER
1) a) x = 2
⎛ 1 ⎞
⎜−1 ⎟
b) i) ⎜ 3 ⎟
⎜ 2 − 1⎟
⎜ ⎟
⎝3 9⎠
ii) h = 7 , k = -4
2) a) k = 4 and t = -8
3
b) x = 5 and y = −
2
3) a) h = 2
⎛3 1 ⎞
⎜ ⎟
b) ⎜ 10 10 ⎟
⎜2 − ⎟
1
⎜ ⎟
⎝5 5⎠
c) x = 2 and y = -1
4) (a) k = -4 , m = 2
3 −3
(b) x = , y=
2 2
−1
5) (a) h = , k=1
3
11 13
(b) x = , y=
3 3

12. ppr maths nbk
⎛ 3 −1 ⎞
6) (a) M = ⎜ ⎟
⎝ 5 −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