a1. (a) det a = 2 (a1) - yr2dpmathssl -...

Matrices (NON GDC) IB Questionbank Maths SL 22

A1. (a) det A = 2 (A1)

A-1 =

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛

121

223

24

13

21 A1 N2 2

(b) evidence of multiplying by A–1 (M1)

e.g. X = A–1⎟⎟⎠

⎞⎜⎜⎝

⎛− 2264

, A–1 B

correct working A1

e.g. X = ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛281020

21,

2264

121

223

,2264

2143

21

X = ⎟⎟⎠

⎞⎜⎜⎝

⎛14510

A2 N3 4

[6]


A2. (a) evidence of considering determinant (M1)

e.g. 3 × –3 – (–2) × x, attempt to find inverse

setting the determinant equal to zero (M1)

e.g. –9 + 2x = 0, 2x = 9

29=x A1 N2 3

(b) METHOD 1

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+−

=−

3 2 3

2911 xx

A (A1) (A1)

Note: Award A1 for Adet1

, A1 for ⎟⎠⎞⎜⎝

⎛ −−3 2

3 x .

one correct equation from A = A–1 (A1)

e.g. xx

xxxx

=+−−−=

+−=

+−=

+−−

29,3

293,2

292,3

293

attempt to solve the equation (M1)

e.g. –3 = 3(–9 + 2x), –9 + 2x = –1

x = 4 (do not accept x = 4, x = 0) A1 N4 5

METHOD 2

A2 = I (A1)

A2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−=

920029x

x (A1)

one correct equation from A2 = I (A1)

e.g. 9 – 2x =1

attempt to solve the equation (M1)

e.g. 2x = 8

x = 4 A1 N4 5 [8]


A3. (a) WP = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

6513

A1A1A1 N3

Note: Award A1 for each correct element.

(b) Note: The first two steps may be done in any order.

subtracting (A1)

e.g. ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101226

– 2WP

multiplying WP by 2 (A1)

e.g. ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

121026

S = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

− 220

A1 N2

[6]


A4. (a) evidence of multiplying (M1) e.g. one correct element

AB = ⎟⎟⎠

⎞⎜⎜⎝

⎛−515

A1A1 N3

(b) METHOD 1

evidence of multiplying by A (on left or right) (M1) e.g. AA–1 X = AB, X = AB

X = ⎟⎟⎠

⎞⎜⎜⎝

⎛−515

(accept x = – 15, y = 5) A1 N2

METHOD 2

attempt to set up a system of equations (M1)

e.g. 5103

,51024

=+−

−=+ yxyx

X = ⎟⎟⎠

⎞⎜⎜⎝

⎛−515

(accept x = – 15, y = 5) A1 N2

[5]


A5. (a) correct substitution into the formula for the determinant (A1) e.g. det A = 9ex × e3x – ex × ex

det A = 9e4x – e2x A1 N2

(b) recognizing that no inverse implies det A = 0 R1 e.g. 9e4x – e2x = 0, ad – bc = 0

attempt to solve equation (M1)

e.g. e2x = 91

, e–2x = 9, e2x(9e2x – 1) = 0, 9e4x = e2x

rearranging to get correct log equation

e.g. 2x = )eln()e9ln(,9ln2,91ln 24 xxx ==− (A1)

isolating x A1

e.g. x 9,21,

31ln,9ln

21,

91ln

21 =−==−= baxx

x = –ln 3 (accept a = –1, b = 3) A1 N3 [7]


A6. (a) (i) AB = ⎟⎟⎠

⎞⎜⎜⎝

⎛4004

(= 4I) A2 N2

(ii) A–1 = ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−

45

23

41

21

,41,

5612

41 B A1 N1

(b) METHOD 1

⎟⎟⎠

⎞⎜⎜⎝

⎛yx

= A–1 C (M1)

=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−48

45

23

41

21

48

5612

41 A1

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛175

yx

A1A1 N3

METHOD 2

5x + y = 8, 6x + 2y = –4 A1 for work towards solving their system (M1)

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛175

yx

A1A1 N3

[7]


A7. (a) METHOD 1

M = (M–1)–1 (M1)

M = ⎟⎟⎠

⎞⎜⎜⎝

⎛− 51

02101

A1A1 N3

METHOD 2

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛1001

2105

dcba

(M1)

5a + b = 1, 2b = 0, 5c + d = 0, 2d = 1 (A1)

M = ⎟⎟⎠

⎞⎜⎜⎝

⎛− 5.01.0

02.0 A1 N3

(b) METHOD 1

evidence of appropriate approach (M1) e.g. X = M–1B

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛71

2105

yx

A1

= ⎟⎟⎠

⎞⎜⎜⎝

⎛155

A1 N2

METHOD 2

evidence of appropriate approach (M1)

e.g. ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛− 7

15.01.002.0

yx

0.2x = 1, –0.1x + 0.5y = 7 A1

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛155

yx

A1 N2

[6]


A8. (a) evidence of addition (M1) e.g. at least two correct elements

A + B = ⎟⎟⎠

⎞⎜⎜⎝

⎛0124

A1 N2

(b) evidence of multiplication (M1) e.g. at least two correct elements

−3A = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−3963

A1 N2

(c) evidence of matrix multiplication (in correct order) (M1)

e.g. AB = ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( )( )⎟⎟⎠⎞

⎜⎜⎝

⎛−+−−++−+

1103213312012231

AB = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−11121

A2 N3

[7]


A9. (a) det M = − 4 A1 N1

(b) M−1 = ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−−

21

21

41

41

2211

41 A1A1 N2

Note: Award A1 for 41− and A1 for the correct

matrix.

(c) X = M−1 ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−−=⎟⎟⎠

⎞⎜⎜⎝

⎛84

2211

41

84

X M1

X = ( )2,323

−==⎟⎟⎠

⎞⎜⎜⎝

⎛−

yx A1A1 N0

Note: Award no marks for an algebraic solution of the system 2x + y = 4, 2x − y = 8.

[6]


A10. (a) evidence of correct method (M1) e.g. at least 1 correct element (must be in a 2 × 2 matrix)

AB = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

++−

−−

236

022ppq

q A1 N2

(b) METHOD 1

evidence of using AB = I (M1) 2 correct equations A1A1

e.g. –2 – 2q = 1 and 3 + 2p

= 1, –6 + pq = 0

p = –4, q = 23− A1A1 N1N1

METHOD 2

finding A–1 = ⎟⎟⎠

⎞⎜⎜⎝

⎛−+ 13

26

1 pp

A1

evidence of using A–1 = B (M2)

e.g. qpp

pqpp

=+

−=+

=+

−=+ 6

3– and 26

,6

3 and 16

2

p = –4, q = 23− A1A1 N1N1

[7]


A11. (a) Attempt to multiply e.g. ⎟⎟⎠

⎞⎜⎜⎝

⎛++−−+90006201

(M1)

A2 = ⎟⎟⎠

⎞⎜⎜⎝

⎛ −9081

A1 N2

(b) 3X + ⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟⎠

⎞⎜⎜⎝

⎛ −1243

3021

(M1)

3X = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−2264

(A1)

X = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−2264

31

A1 N2

[5]


A12. ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−0000

1001

4312

64312

4312

k (A1)

M2 = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−191867

A2

6M = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−2418612

A1

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛−

−0000

00

5005

kk

A1

k = 5 A1 N2 [6]


A13. (a) det A = 5 (A1)

A–1 = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−7283

51

A1 N2

(b) Set up matrix equation ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛11

3287yx

(M1)

premultiplying by A–1 M1

A–1⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −

11

3287 1Ayx

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟⎠

⎞⎜⎜⎝

⎛11

55

51

yx

yx

A1

x = –1, y = 1 A1 N0 [6]


A14. (a) (i) a = 5 A1 N1

(ii) b + 9 = 4 (M1)

b = −5 A1 N2

(b) Comparing elements 3(2) − 5(q) = −9 M1

q = 3 A2 N2 [6]


A15. (a) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

8246

2k

A (A1)

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=−51224

2k

BA A2 N3

(b) Evidence of using the definition of determinant (M1) Correct substitution (A1) eg 4(5) − 2(2k − 1), 20 − 2(2k − 1), 20 − 4k + 2 det (2A − B) = 22 − 4k A1 N3

[6]


A16. (a) A + B = ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛ed

bca 01

0

= ⎟⎟⎠

⎞⎜⎜⎝

⎛++

eb

dca 1

A2 2

(b) AB = ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛ed

bca 01

0 A1A1A1A1 4

Note: Award N2 for finding BA = ⎟⎟⎠

⎞⎜⎜⎝

⎛+ bd

bcead

a.

[6]


A17. (a) 4 8 5 2

32 14 1 a−⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠Q (A1)

9 63

3 14 a−⎛ ⎞

= ⎜ ⎟−⎝ ⎠Q (A1)

3 21413a

−⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟⎝ ⎠

Q (A1) (N3) 3

(b) 2 4 5 21 7 1 a−⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠CD

14 4 42 2 7

aa

− − +⎛ ⎞= ⎜ ⎟− +⎝ ⎠

(A1)(A1)(A1)(A1) (N4) 4

(c) det 5 2a= +D (may be implied) (A1)

1 211 55 2a

a− −⎛ ⎞= ⎜ ⎟+ ⎝ ⎠

D (A1) (N2) 2

[9]


A18. ⎟⎟⎠

⎞⎜⎜⎝

⎛65–13

X + ⎟⎟⎠

⎞⎜⎜⎝

⎛1001

X = ⎟⎟⎠

⎞⎜⎜⎝

⎛3–084

⎟⎟⎠

⎞⎜⎜⎝

⎛75–14

X = ⎟⎟⎠

⎞⎜⎜⎝

⎛3–084

(M1)

Pre-multiply by inverse of ⎟⎟⎠

⎞⎜⎜⎝

⎛75–14

(M1)

X = ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛3–084

451–7

331

(A1)(A1)

Note: Award (A1) for determinant, (A1) for matrix ⎟⎟⎠

⎞⎜⎜⎝

⎛451–7

.

= ⎟⎟⎠

⎞⎜⎜⎝

⎛28205928

331

(A1)(A1)(A1)(A1)

⎟⎠⎞⎜

⎝⎛ ====⇒

3328,

3320,

3359,

3328 dcba

OR

⎟⎟⎠

⎞⎜⎜⎝

⎛65–13

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛dcba

dcba

⎟⎟⎠

⎞⎜⎜⎝

⎛3–084

(A1)

⎟⎟⎠

⎞⎜⎜⎝

⎛++++dbcadbca65–65–

33+ ⎟⎟⎠

⎞⎜⎜⎝

⎛dcba

= ⎟⎟⎠

⎞⎜⎜⎝

⎛3–084

(A1)

4a + c = 4 –5a + 7c = 0 (A1) 4b + d = 8 –5b + 7d = –3 (A1)

Notes: Award (A1) for each pair of equations. Allow ft from their equations.

a = 3328

, b = 3359

, c = 3320

, d = 3328

(A1)(A1)(A1)(A1)

Note: Award (A0)(A0)(A1)(A1) if the final answers are given as decimals ie 0.848, 1.79, 0.606, 0.848.

[8]


A19. 2p2 + 12p = 14 (M1) (A1) p2 + 6p – 7 = 0 (p + 7)(p – 1) = 0 (A1) p = –7 or p = 1 (A1) (C4)

Note: Both answers are required for the final (A1). [4]


A20. (a) M2 = ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛− 12

2122 aa

= ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+5222242

aaa (A1)(A1)(A1)(A1) 4

(b) 2a – 2 = –4 ⇒ a = –1 (A1) Substituting: a2 + 4 = (–1)2 + 4 = 5 (A1) 2

Note: Candidates may solve a2 + 4 = 5 to give a = ±1, and then show that only a = –1 satisfies 2a – 2 = –4.

(c) M = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−1221

M–1 = – ⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−1221

31 (M1)

= ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

31

32

32

31

or 1221

31 (A1)

–x + 2y = –3 2x – y = 3

⇒ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

−33

1221

yx

(M1)(M1)

⇒ ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛33

1221

31

32

32

31

31

32

32

31

yx

(A1)

⇒ ⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛11

yx

(A1) 6

ie x = 1 y = –1

Note: The solution must use matrices. Award no marks for solutions using other methods.

[12]


A21. B = (BA)A–1 (M1)

= – ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛5220

844211

41 (M1)

= – ⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−4816124

41 (A1)

= ⎟⎟⎠

⎞⎜⎜⎝

⎛12431

(A1)

OR

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛844211

0225

dcba

(M1)

⇒ ⎭⎬⎫

==+221125

aba

⇒ a = 1, b = 3 (A1)

⎭⎬⎫

==+824425

cdc

⇒ c = 4, d = 12 (A1)

B = ⎟⎟⎠

⎞⎜⎜⎝

⎛12431

(A1) (C4)

Note: Correct solution with inversion (ie AB instead of BA) earns FT marks, (maximum [3 marks]).

[4]

Matrices (GDC OPTIONAL) IB Questionbank Maths SL 1

1. Let A = ⎟⎟⎠

⎞⎜⎜⎝

⎛ −41311 x

and B = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

2

3x .

(a) Find AB.

(b) The matrix C = ⎟⎟⎠

⎞⎜⎜⎝

⎛2820

and 2AB = C. Find the value of x.

(Total 6 marks)


2. Let A = ⎟⎟⎠

⎞⎜⎜⎝

⎛0220

.

(a) Find

(i) A−1;

(ii) A2. (4)

Let B = ⎟⎟⎠

⎞⎜⎜⎝

⎛q

p02

.

(b) Given that 2A + B = ⎟⎟⎠

⎞⎜⎜⎝

⎛3462

, find the value of p and of q.

(3)

(c) Hence find A−1B. (2)

(d) Let X be a 2 × 2 matrix such that AX = B. Find X. (2)

(Total 11 marks)


3. Let Sn be the sum of the first n terms of the arithmetic series 2 + 4 + 6 + ….

(a) Find

(i) S4;

(ii) S100. (4)

Let M = ⎟⎟⎠

⎞⎜⎜⎝

⎛1021

.

(b) (i) Find M2.

(ii) Show that M3 = ⎟⎟⎠

⎞⎜⎜⎝

⎛1061

.

(5)

It may now be assumed that Mn = ⎟⎟⎠

⎞⎜⎜⎝

⎛1021 n

, for n ≥ 4. The sum Tn is defined by

Tn = M1 + M2 + M3 + ... + Mn .

(c) (i) Write down M4.

(ii) Find T4. (4)

(d) Using your results from part (a) (ii), find T100. (3)

(Total 16 marks)


4. Matrices A, B and C are defined by

A = ⎟⎟⎠

⎞⎜⎜⎝

⎛2715

B = ⎟⎟⎠

⎞⎜⎜⎝

⎛− 153

42 C = ⎟⎟⎠

⎞⎜⎜⎝

⎛ −2879

.

Let X be an unknown 2 × 2 matrix satisfying the equation

AX + B = C.

This equation may be solved for X by rewriting it in the form

X = A−1 D.

where D is a 2 × 2 matrix.

(a) Write down A−1. (2)

(b) Find D. (3)

(c) Find X. (2)

(Total 7 marks)


5. Consider the matrix A = ⎟⎟⎠

⎞⎜⎜⎝

⎛172–5

.

(a) Write down the inverse, A–l. (2)

(b) B, C and X are also 2 × 2 matrices.

(i) Given that XA + B = C, express X in terms of A–1, B and C.

(ii) Given that B = ⎟⎟⎠

⎞⎜⎜⎝

⎛2–576

, and C = ,78–05–⎟⎟⎠

⎞⎜⎜⎝

⎛ find X.

(4) (Total 6 marks)


A1. (a) Attempting to multiply matrices (M1)

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++−+=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −xx

xx

xx

171

8923

2

3

41311 22

A1A1 N3

(b) Setting up equation M1

eg ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛+

+⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛+

+1410

171,

2820

23422,

2820

1712

222

xx

xx

xx

⎟⎟⎠

⎞⎜⎜⎝

⎛=+=+

=+=+

1417101

2823420 2 2 22

xx

xx (A1)

x = −3 A1 N2 [6]


A2. (a) (i) A−1 = ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

021

210

A2 N2

(ii) A2 = ⎟⎟⎠

⎞⎜⎜⎝

⎛4004

A2 N2

(b) ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛3462

02

0440

qp

(M1)

p = 2, q = 3 A1A1 N3

(c) Evidence of attempt to multiply (M1)

eg A−1B = ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

021

210

⎟⎟⎠

⎞⎜⎜⎝

⎛3022

A−1B = ⎟⎟

⎠

⎞⎜⎜

⎝

⎛

11230

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

121

210

acceptp

q A1 N2

(d) Evidence of correct approach (M1)

eg X = A−1B, setting up a system of equations

X = ⎟⎟

⎠

⎞⎜⎜

⎝

⎛

11230

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

121

210

acceptp

q A1 N2

[11]


A3. (a) (i) S4 = 20 A1 N1

(ii) u1 = 2, d = 2 (A1)

Attempting to use formula for Sn M1

S100 = 10100 A1 N2

(b) (i) M2 = ⎟⎟⎠

⎞⎜⎜⎝

⎛1041

A2 N2

(ii) For writing M3 as M2 × M or M × M2 ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛1041

1021

or M1

M3 = ⎟⎟⎠

⎞⎜⎜⎝

⎛++++10002401

A2

M3 = ⎟⎟⎠

⎞⎜⎜⎝

⎛1061

AG N0

(c) (i) M4 = ⎟⎟⎠

⎞⎜⎜⎝

⎛1081

A1 N1

(ii) T4 = ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛1081

1061

1041

1021

(M1)

= ⎟⎟⎠

⎞⎜⎜⎝

⎛40204

A1A1 N3

(d) T100 = ⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛10

2001...

1041

1021

(M1)

= ⎟⎟⎠

⎞⎜⎜⎝

⎛1000

10100100 A1A1 N3

[16]


A4. (a) A−1 = ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−

35

37

31

32

or ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−67.133.2333.0667.0

or5712

31 (A1)(A1) (N2)

(b) AX = C − B (may be implied) (A1)

X = A−1 (C−B) (A1)

D = C − B

= ⎟⎟⎠

⎞⎜⎜⎝

⎛−−1311117

(A1) (N3)

(c) X = ⎟⎟⎠

⎞⎜⎜⎝

⎛ −4231

(A2) (N2)

[7]


A5. (a) det A = 5(1) – 7(–2) = 19

A–1 = ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−

195

197192

191

5721

191

(A2)

Note: Award (A1) for ⎟⎟⎠

⎞⎜⎜⎝

⎛− 57

21, (A1) for dividing by 19.

OR

A–1 = ⎟⎟⎠

⎞⎜⎜⎝

⎛− 263.0368.0

105.00526.0 (G2) 2

(b) (i) XA + B = C ⇒ XA = C – Β (M1) X = (C – Β)Α–1 (A1)

OR

X = (C – B)A–1 (A2)

(ii) (C – Β)Α–1 = ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−

195

197192

191

913711

(A1)

⇒ X = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−

1432

1919

1976

1957

1938

(A1)

OR

X = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−1432

(G2) 4

Note: If premultiplication by A–1 is used, award (M1)(M0) in

part (i) but award (A2) for ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ −

1994

1912

1911

1937

in part (ii).

[6]

Matrices (GDC) IB Questionbank Maths SL 1

1. The system of linear equations below can be written as the matrix equation MX = N.

x + 6y – 3z = –1 4x + 2y – 4z = 12 x + y + 5z = 15

(a) Write down the matrices M and N. (3)

(b) Solve the matrix equation MX = N. (3)

(c) Hence write down the solution of the system of linear equations. (1)

(Total 7 marks)


2. Let A = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

−

342411321�and B =

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−132

.

(a) Write down A–1. (2)

(b) Solve AX = B. (3)

(Total 5 marks)


3. Consider the function f(x) = px3 + qx2 + rx. Part of the graph of f is shown below.

The graph passes through the origin O and the points A(–2, –8), B(1, –2) and C(2, 0).

(a) Find three linear equations in p, q and r. (4)

(b) Hence find the value of p, of q and of r. (3)

(Total 7 marks)


4. Let f(x) = ax2 + bx + c where a, b and c are rational numbers.

(a) The point P(–4, 3) lies on the curve of f. Show that 16a –4b + c = 3. (2)

(b) The points Q(6, 3) and R(–2, –1) also lie on the curve of f. Write down two other linear equations in a, b and c.

(2)

(c) These three equations may be written as a matrix equation in the form AX = B,

where X = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

cba

.

(i) Write down the matrices A and B.

(ii) Write down A–1.

(iii) Hence or otherwise, find f(x). (8)

(d) Write f(x) in the form f(x) = a(x – h)2 + k, where a, h and k are rational numbers. (3)

(Total 15 marks)


5. Let A = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−

124032103

.


(b) Let B be a 3 × 3 matrix. Given that AB + ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

571856767

1029435123

, find B.

(4) (Total 6 marks)


6. Let A = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

220112311

.


The matrix B satisfies the equation 1

21 −

⎟⎠⎞⎜

⎝⎛ − BI = A, where I is the 3 × 3 identity matrix.

(b) (i) Show that B = –2(A–1 – I).

(ii) Find B.

(iii) Write down det B.

(iv) Hence, explain why B–1 exists. (6)

Let BX = C, where X = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

zyx�and C =

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−112

.

(c) (i) Find X.

(ii) Write down a system of equations whose solution is represented by X. (5)

(Total 13 marks)


7. (a) Write down the inverse of the matrix A = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−

−

351122131

.

(2)

(b) Hence solve the simultaneous equations

x – 3y + z = 1 2x + 2y – z = 2 x – 5y + 3z = 3

(4) (Total 6 marks)



⎠

⎞

⎜⎜⎜

⎝

⎛ −

314102031

.

(b) Hence or otherwise solve

x − 3y = 1

2x + z = 2

4x + y + 3z = −1

(Total 6 marks)


9. Let A = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

102213321

, B = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

132318

and X = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

zyx

.

(a) Write down the inverse matrix A−1.

(b) Consider the equation AX = B.

(i) Express X in terms of A−1 and B.

(ii) Hence, solve for X. (Total 6 marks)


10. The matrix A = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−122113021

has inverse A−1 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −−−

ba 6113221

.

(a) Write down the value of

(i) a;

(ii) b.

Consider the simultaneous equations

x + 2y = 7

–3x + y – z = 10

2x – 2y + z = –12

(b) Write these equations as a matrix equation.

(c) Solve the matrix equation. (Total 6 marks)


11. The function f is given by f (x) = mx3 + nx2 + px + q, where m, n, p, q are integers. The graph of f passes through the point (0, 0).

(a) Write down the value of q. (1)

The graph of f also passes through the point (3, 18).

(b) Show that 27 m+ 9n + 3p =18.

The graph of f also passes through the points (1, 0) and (–1, –10). (2)

(c) Write down the other two linear equations in m, n and p. (2)

(d) (i) Write down these three equations as a matrix equation.

(ii) Solve this matrix equation. (6)

(e) The function f can also be written f (x) = x (x −1)(rx − s) where r and s are integers. Find r and s.

(3) (Total 14 marks)



⎠

⎞

⎜⎜⎜

⎝

⎛

−−

−

351122131

(b) Hence solve the simultaneous equations

x – 3y + z = 1

2x + 2y – z = 2

x – 5y + 3z = 3 (Total 6 marks)


A1. (a) M = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−

15121

,511424361N A2A1 N3 3

(b) evidence of appropriate approach (M2)

e.g. X = M–1N, attempting to solve a system of three equations

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=205

X A1 N3 3

(c) x = 5, y = 0, z = 2 A1 N1 1 [7]


A2. (a) A–1 =

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−

−−

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−

−−

310

32

311

35

352

313

333.00667.0333.0167.167.1233.4

A2 N2

(b) evidence of attempting to solve equation (M1) e.g. multiply by A–1 (on left or right), setting up system of equations

X = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

101

(accept x = 1, y = 0, z = –1) A2 N3

[5]


A3. (a) attempt to substitute points into the function (M1) e.g. –8 = p(–2)3 + q(–2)2 + r(–2), one correct equation

–8 = –8p + 4q – 2r, –2 = p + q + r, 0 = 8p + 4q + 2r A1A1A1 N4

(b) attempt to solve system (M1) e.g. inverse of a matrix, substitution

p = 1, q = –1, r = –2 A2 N3

Notes: Award A1 for two correct values. If no working shown, award N0 for two correct values.

[7]


A4. (a) evidence of substituting (–4, 3) (M1) correct substitution 3 = a(–4)2 + b(–4) + c A1 16a – 4b + c = 3 AG N0

(b) 3 = 36a + 6b + c, –1 = 4a – 2b + c A1A1 N1N1

(c) (i) A = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

133

;12416361416B A1A1 N1N1

(ii) A–1 =

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−

−

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−

−

23

101

53

81

403

51

161

801

201

5.11.06.0125.0075.02.00625.00125.005.0

A2 N2

(iii) evidence of appropriate method (M1) e.g. X = A–1B, attempting to solve a system of three equations

X = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−35.025.0

(accept fractions) A2

f(x) = 0.25x2 – 0.5x – 3 (accept a = 0.25, b = –0.5, c = –3, or fractions) A1 N2

(d) f(x) = 0.25(x – 1)2 – 3.25 (accept h = 1, k = –3.25, a = 0.25, or fractions) A1A1A1 N3 [15]


A5. (a) A–1 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−

968212323

A2 N2

(b) evidence of subtracting matrices (M1)

e.g. ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

1551012218410

,1029435123

571856767

, D – C

evidence of multiplying on left by A–1 (M1)

e.g. A–1 AB, A–1(D – C), ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−

1551012218410

968212323

B = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

114201312

A2 N3

[6]


A6. (a) A–1 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−

−−

75.05.0125.15.01111

A2 N2

(b) (i) I – 21

B = A–1 A1

21− B = A–1 – I A1

B = –2(A–1 – I) AG

(ii) B = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

−

5.0125.232224

A2 N2

(iii) det B = 12 A1 N1

(iv) det B ≠ 0 R1 N1

(c) (i) evidence of using a valid approach M1 e.g. X = B–1C

X =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

34131

33.11333.0

A1 N1

(ii) 4x – 2y + 2z = 2, –2x + 3y – 2.5z = –1, –2x + y + 0.5z = 1 A1A1A1 N3 [13]


A7. (a) A–1 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−

8.02.02.13.02.07.01.04.01.0

A2 N2

(b) For recognizing that the equations may be written as A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

zyx

(M1)

for attempting to calculate ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

6.16.02.1

321

1Azyx

M1

x = 1.2, y = 0.6, z = 1.6 (accept row or column vectors) A2 N3 [6]


A8. (a) A−1 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−

2.16.24.02.06.04.06.08.12.0

A2 N2

(b) For recognizing that the equations may be written as A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

121

zyx

(M1)

For attempting to calculate ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

121

A 1

zyx

(M1)

x = 4, y = 1, z = −6 A2 N4 [6]


A9. (a) A−1 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−−

−−

1.6733.1667.02.3367.1333.00.3330.667333.0

or

35

34

32

37

35

31

31

32

31

A2 N2

(b) (i) X = A−1B A1 N1

(ii) X = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

325

A3 N3

[6]


A10. (a) (i) a = 4 A1 N1

(ii) b = 7 A1 N1

(b) EITHER

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

12107

zyx

A1 N1

OR

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

12107

122113021

zyx

A1 N1

(c) ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

12107

1Azyx

(accept algebraic method) (M1)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

453

zyx

(accept x = −3, y = 5, z = 4) A2 N3

[6]


A11. (a) q = 0 A1 N1

(b) Attempting to substitute (3, 18) (M1)

m33 + n32 + p3 = 18 A1 27m + 9n + 3p = 18 AG N0

(c) m + n + p = 0 A1 N1

− m + n − p = −10 A1 N1

(d) (i) Evidence of attempting to set up a matrix equation (M1)

Correct matrix equation representing the given equations A2 N3

eg ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−− 10018

1111113927

pnm

(ii) ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−352

A1A1A1 N3

(e) Factorizing (M1)

eg f (x) = x(2x2 − 5x + 3), f (x) = (x2 − x)(rx − s)

r = 2 s = 3 (accept f (x) = x(x − 1)(2x − 3)) A1A1 N3 [14]


A12. (a) ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−

8.03.01.0

2.02.04.0

2.17.01.0

A2 3

(b) For recognizing that the equations may be written as A ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

zyx

(M1)

for attempting to calculate ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

6.16.02.1

321

1Azyx

M1

x = 1.2, y = 0.6, z = 1.6 (Accept row or column vectors) A2 3 [6]

a1. (a) det a = 2 (a1) - yr2dpmathssl -...

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