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Date	May 2019	Marks available	6	Reference code	19M.2.AHL.TZ0.F_3
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term		Question number	F_3	Adapted from	N/A

Question

The matrix M is given by M $= \left[ {\begin{array}{*{20}{c}} 1&2&2 \\ 3&1&1 \\ 2&3&1 \end{array}} \right]$ .

Given that M³ can be written as a quadratic expression in M in the form aM² + bM + cI , determine the values of the constants a, b and c.

[7]

a.

Show that M⁴ = 19M² + 40M + 30I.

[2]

b.

Using mathematical induction, prove that Mⁿ can be written as a quadratic expression in M for all positive integers n ≥ 3.

[6]

c.

Find a quadratic expression in M for the inverse matrix M^–1.

[2]

d.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1

M² $= \left[ {\begin{array}{*{20}{c}} {11}&{10}&6 \\ 8&{10}&8 \\ {13}&{10}&8 \end{array}} \right]$ ; M³ $= \left[ {\begin{array}{*{20}{c}} {53}&{50}&{38} \\ {54}&{50}&{34} \\ {59}&{60}&{44} \end{array}} \right]$ (A1)(A1)

let $\left[ {\begin{array}{*{20}{c}} {53}&{50}&{38} \\ {54}&{50}&{34} \\ {59}&{60}&{44} \end{array}} \right] = a\left[ {\begin{array}{*{20}{c}} {11}&{10}&6 \\ 8&{10}&8 \\ {13}&{10}&8 \end{array}} \right] + b\left[ {\begin{array}{*{20}{c}} 1&2&2 \\ 3&1&1 \\ 2&3&1 \end{array}} \right] + c\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right]$ (M1)

then, for example

11a + b + c = 53

10a + 2b = 50 M1A1

6a + 2b = 38

the solution is a = 3, b =10, c =10 (M1)A1

(M³ = 3M² +10M +10I)

METHOD 2

det(M − $\lambda$ I) = 0 (M1)

$\Rightarrow \left( {1 - \lambda } \right)\left( {{{\left( {1 - \lambda } \right)}^2} - 3} \right) - 2\left( {3\left( {1 - \lambda } \right) - 2} \right) + 2\left( {9 - 2\left( {1 - \lambda } \right)} \right) = 0$ M1A1

$- {\lambda ^3} + 3{\lambda ^2} + 10\lambda + 10 = 0$ M1A1

applying the Cayley – Hamilton theorem (M1)

M³ = 3M² +10M +10I and so a = 3, b =10, c =10 A1

[7 marks]

a.

M⁴ = 3M³ + 10M² + 10M M1

= 3(3M² + 10M + 10I) + 10M² + 10M M1

=19M² + 40M +30I AG

[2 marks]

b.

the statement is true for n = 3 as shown in part (a) A1

assume true for n = k, ie M^k = pM² + qM + rI M1

Note: Subsequent marks after this M1 are independent and can be awarded.

M^k⁺¹ = pM³ + qM² + rM M1

= p(3M² +10M +10I) + qM² + rM M1

= (3p +q)M² + (10p + r)M + 10pI A1

hence true for n = k ⇒ true for n = k +1 and since true for n = 3, the statement is proved by induction R1

Note: Award R1 provided at least four of the previous marks are gained.

[6 marks]

c.

M² = 3M + 10I + 10M^–1 M1

M^–1 = $\frac{1}{{10}}$ (M² − 3M − 10I) A1

[2 marks]

d.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.

Syllabus sections

Topic 1—Number and algebra » AHL 1.14—Introduction to matrices

Show 139 related questions

22M.2.AHL.TZ1.7a.i:
By considering the image of $(0, 0)$ , find $q$ .
22M.2.AHL.TZ1.7c.ii:
Hence find the angle and direction of the rotation represented by $R$ .
22M.2.AHL.TZ1.7b:
Write down the matrix $S$ .
22M.1.AHL.TZ2.15b:
Find the value of $det M$ .
EXM.1.AHL.TZ0.26b:
Calculate det (BA).
EXM.1.AHL.TZ0.6a:
Find Q.
EXM.2.AHL.TZ0.23a:
Find the values of $\lambda$ for which the matrix (A − $\lambda$ I) is singular.
EXM.1.AHL.TZ0.7b:
Let $3\left( {\begin{array}{*{20}{c}} { - 4}&8 \\ 2&1 \end{array}} \right) - 5\left( {\begin{array}{*{20}{c}} 2&0 \\ q&{ - 4} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 22}&{24} \\ 9&{23} \end{array}} \right)$ .

Find the value of $q$ .
22M.2.AHL.TZ1.7a.ii:
By considering the image of $(1, 0)$ and $(0, 1)$ , show that

$P = (\begin{matrix} \frac{\sqrt{3}}{4} & \frac{1}{4} \\ - \frac{1}{4} & \frac{\sqrt{3}}{4} \end{matrix})$ .
22M.2.AHL.TZ1.7d.ii:
Find the value of $a$ and the value of $b$ .
EXM.2.AHL.TZ0.4:
The matrices A, B, X are given by

A = $\left( {\begin{array}{*{20}{c}} 3&1 \\ { - 5}&6 \end{array}} \right)$ , B = $\left( {\begin{array}{*{20}{c}} 4&8 \\ 0&{ - 3} \end{array}} \right)$ , X = $\left( {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right)$ , ${\text{where}}$ $a$ , $b$ , $c$ , $d \in \mathbb{Q}$ .

Given that AX + X = Β, find the exact values of $a$ , $b$ , $c$ and $d$ .
EXM.2.AHL.TZ0.8d.i:
Write down these three equations as a matrix equation.
21N.2.AHL.TZ0.4b:
Three drones are initially positioned at the points $A$ , $B$ and $C$ . After performing the movements listed above, the drones are positioned at points $A'$ , $B'$ and $C'$ respectively.

Show that the area of triangle $ABC$ is equal to the area of triangle $A' B' C'$ .
21N.2.AHL.TZ0.4a.iii:
Find $P^{2}$ .
21N.2.AHL.TZ0.4c:
Find a single transformation that is equivalent to the three transformations represented by matrix $P$ .
22M.2.AHL.TZ1.7d.i:
Write down an equation satisfied by $(\begin{matrix} a \\ b \end{matrix})$ .
22M.2.AHL.TZ1.7c.i:
Use $P = R S$ to find the matrix $R$ .
21N.2.AHL.TZ0.4a.i:
Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.
21N.2.AHL.TZ0.4a.ii:
Find a single matrix $P$ that defines a transformation that represents the overall change in position.
21N.2.AHL.TZ0.4a.iv:
Hence state what the value of $P^{2}$ indicates for the possible movement of the drone.
EXM.1.AHL.TZ0.30c:
Find the matrix X, such that AX = C.
EXM.2.AHL.TZ0.8a:
Write down the value of $q$ .
EXM.1.AHL.TZ0.27a:
(A + B)² = A² + 2AB + B²
EXM.2.AHL.TZ0.23b.iii:
Use the result from part (b) (ii) to explain why A is non-singular.
EXM.2.AHL.TZ0.10c:
Hence find A⁻¹B.
EXM.1.AHL.TZ0.27c:
CA = B C = $\frac{B}{A}$
EXM.1.AHL.TZ0.34:
The square matrix X is such that X³ = 0. Show that the inverse of the matrix (I – X) is I + X + X².
18M.1.AHL.TZ0.F_2a:
Show that A⁴ = 12A + 5I.
EXM.1.AHL.TZ0.7a.ii:
Find the value of $b$ .
19M.2.AHL.TZ0.F_3a:
Given that M³ can be written as a quadratic expression in M in the form aM² + bM + cI , determine the values of the constants a, b and c.
19M.2.AHL.TZ0.F_3d:
Find a quadratic expression in M for the inverse matrix M^–1.
EXM.2.AHL.TZ0.6a:
Write down A⁻¹.
EXM.2.AHL.TZ0.6b:
Find D.
EXM.2.AHL.TZ0.7c.ii:
Find T₄.
EXM.1.AHL.TZ0.45b.i:
Find the matrix DA.
EXM.1.AHL.TZ0.8a:
Write down the inverse matrix A⁻¹.
EXM.2.AHL.TZ0.7d:
Using your results from part (a) (ii), find T₁₀₀.
EXM.1.AHL.TZ0.9a.ii:
Write down the value of $b$ .
EXM.2.AHL.TZ0.5b:
find AB.
EXM.2.AHL.TZ0.7a.i:
Find $S$ ₄.
EXM.1.AHL.TZ0.9a.i:
Write down the value of $a$ .
EXM.2.AHL.TZ0.5a:
write down A + B.
EXM.1.AHL.TZ0.9b:
Write these equations as a matrix equation.
EXM.1.AHL.TZ0.8b.i:
Express X in terms of A⁻¹ and B.
EXM.2.AHL.TZ0.11b:
Write down M⁻¹.
EXM.2.AHL.TZ0.11a:
Write down the determinant of M.
EXM.2.AHL.TZ0.20c.ii:
A^–1 = I – A.
EXM.1.AHL.TZ0.54a:
Find det A.
EXM.1.AHL.TZ0.27b:
(A – kI)³ = A³ – 3kA² + 3k²A – k³
EXM.2.AHL.TZ0.8b:
Show that $27m + 9n + 3p = 18$ .
EXM.2.AHL.TZ0.3c:
Using this value of $a$ , find ${M^{ - 1}}$ and hence solve the system of equations:

$- x + 2y = - 3$

$2x - y = 3$
EXM.2.AHL.TZ0.20c.i:
A³ = –I.
EXM.2.AHL.TZ0.6c:
Find X.
EXM.2.AHL.TZ0.20a.ii:
Show that ${\gamma ^2} = \gamma - 1$ .
EXM.1.AHL.TZ0.11a:
Find A + B.
EXM.2.AHL.TZ0.7b.i:
Find M².
EXM.1.AHL.TZ0.16b.ii:
Write down ${M^{ - 1}}$ for this value of $a$ .
EXM.1.AHL.TZ0.4b.i:
Given that XA + B = C, express X in terms of A^–1, B and C.
EXM.1.AHL.TZ0.46:
If A = $\left( {\begin{array}{*{20}{c}} x&4 \\ 4&2 \end{array}} \right)$ and B = $\left( {\begin{array}{*{20}{c}} 2&y \\ 8&4 \end{array}} \right)$ , find 2 values of $x$ and $y$ , given that AB = BA.
EXM.2.AHL.TZ0.20a.iii:
Hence find the value of ${\left( {1 - \gamma } \right)^6}$ .
EXM.2.AHL.TZ0.3a:
Find ${M^2}$ in terms of $a$ .
EXM.1.AHL.TZ0.45b.ii:
Find B if AB = C.
EXM.2.AHL.TZ0.22c:
If all of the elements of M are positive, find the range of possible values for $a$ .
19M.2.AHL.TZ0.F_3b:
Show that M⁴ = 19M² + 40M + 30I.
EXM.1.AHL.TZ0.9c:
Solve the matrix equation.
EXM.1.AHL.TZ0.16a:
Find a relationship between $a$ and $b$ if the matrices $M = \left( {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right)$ and $N = \left( {\begin{array}{*{20}{c}} 1&b \\ 2&3 \end{array}} \right)$ commute under matrix multiplication.
19M.1.AHL.TZ0.F_13b:
It is now given that A is singular.

By considering appropriate determinants, prove that f is not a bijection.
EXM.1.AHL.TZ0.3:
$A$ and $B$ are 2 × 2 matrices, where $A = \left[ {\begin{array}{*{20}{c}} 5&2 \\ 2&0 \end{array}} \right]$ and $BA = \left[ {\begin{array}{*{20}{c}} {11}&2 \\ {44}&8 \end{array}} \right]$ . Find $B$
EXM.1.AHL.TZ0.29b:
Calculate M².
EXM.1.AHL.TZ0.7a.i:
Write down the value of $a$ .
19M.1.AHL.TZ0.F_13a:
Given that A is non-singular, prove that f is a bijection.
EXM.2.AHL.TZ0.21a.ii:
does X^–1 + Y^–1 have an inverse? Justify your conclusion.
EXM.2.AHL.TZ0.9b:
Hence or otherwise solve

$x - 3y = 1$

$2x + z = 2$

$4x + y + 3z = - 1$
EXM.2.AHL.TZ0.8e:
The function $f$ can also be written $f(x) = x\left( {x - 1} \right)\left( {rx - s} \right)$ where $r$ and $s$ are integers. Find $r$ and $s$ .
EXM.1.AHL.TZ0.16b.i:
Find the value of $a$ if the determinant of matrix $M$ is −1.
EXM.2.AHL.TZ0.21a.i:
find X and Y.
EXM.2.AHL.TZ0.7b.ii:
Show that M³ = $\left( {\begin{array}{*{20}{c}} 1&6 \\ 0&1 \end{array}} \right)$ .
EXM.1.AHL.TZ0.31:
Find the determinant of A, where A = $\left( {\begin{array}{*{20}{c}} 3&1&2 \\ 9&5&8 \\ 7&4&6 \end{array}} \right)$ .
EXM.2.AHL.TZ0.7c.i:
Write down M⁴.
EXM.1.AHL.TZ0.35b:
Hence, find the point of intersection of the three planes.

$\begin{array}{*{20}{c}} {x - 3y + z = 1} \\ {2x + 2y - z = 2} \\ {x - 5y + 3z = 3} \end{array}$
EXM.2.AHL.TZ0.21b.ii:
and hence find an expression for ${A^n} + {\left( {{A^n}} \right)^{ - 1}}$ .
EXM.1.AHL.TZ0.28:
Consider the matrix A $= \left( {\begin{array}{*{20}{c}} {{{\text{e}}^x}}&{{{\text{e}}^{ - x}}} \\ {2 + {{\text{e}}^x}}&1 \end{array}} \right)$ , where $x \in \mathbb{R}$ .

Find the value of $x$ for which A is singular.
EXM.2.AHL.TZ0.11c:
Hence solve M $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 4 \\ 8 \end{array}} \right)$ .
EXM.2.AHL.TZ0.9a:
Write down the inverse of the matrix A = $\left( {\begin{array}{*{20}{c}} 1&{ - 3}&0 \\ 2&0&1 \\ 4&1&3 \end{array}} \right)$ .
EXM.1.AHL.TZ0.8b.ii:
Hence, solve for X.
EXM.1.AHL.TZ0.29a:
Show that M is non-singular.
EXM.2.AHL.TZ0.10b:
Given that 2A + B = $\left( {\begin{array}{*{20}{c}} 2&6 \\ 4&3 \end{array}} \right)$ , find the value of $p$ and of $q$ .
EXM.2.AHL.TZ0.23c:
Use the values from part (b) (i) to express A⁴ in the form $p$ A+ $q$ I where $p$ , $q \in \mathbb{Z}$ .
EXM.1.AHL.TZ0.54b:
Find the set of values of $k$ for which the system has a unique solution.
EXM.1.AHL.TZ0.45a:
Given matrices A, B, C for which AB = C and det A ≠ 0, express B in terms of A and C.
EXM.1.AHL.TZ0.45c:
Find the coordinates of the point of intersection of the planes $x + 2y + 3z = 5$ , $2x - y + 2z = 7$ , $3x - 3y + 2z = 10$ .
EXM.2.AHL.TZ0.8c:
Write down the other two linear equations in $m$ , $n$ and $p$ .
EXM.1.AHL.TZ0.30b:
Hence, or otherwise, find A^–1.
EXM.1.AHL.TZ0.43a:
Find the values of $a$ and $b$ given that the matrix $A = \left( {\begin{array}{*{20}{c}} a&{ - 4}&{ - 6} \\ { - 8}&5&7 \\ { - 5}&3&4 \end{array}} \right)$ is the inverse of the matrix $B = \left( {\begin{array}{*{20}{c}} 1&2&{ - 2} \\ 3&b&1 \\ { - 1}&1&{ - 3} \end{array}} \right)$ .
20N.1.AHL.TZ0.F_4b.ii:
Assuming that $A$ is non-singular, use the result in part (b)(i) to show that

$A^{- 1} = \frac{1}{β} (α I - A)$ .
EXM.2.AHL.TZ0.23b.i:
Find the value of $m$ and of $n$ .
EXM.1.AHL.TZ0.4b.ii:
Given that B $= \left( {\begin{array}{*{20}{c}} 6&7 \\ 5&{ - 2} \end{array}} \right)$ , and C $= \left( {\begin{array}{*{20}{c}} { - 5}&0 \\ { - 8}&7 \end{array}} \right)$ , find X.
EXM.2.AHL.TZ0.10a.i:
Find A⁻¹.
EXM.1.AHL.TZ0.22b:
The matrix C = $\left( {\begin{array}{*{20}{c}} {20} \\ {28} \end{array}} \right)$ and 2AB = C. Find the value of $x$ .
EXM.1.AHL.TZ0.43b:
For the values of $a$ and $b$ found in part (a), solve the system of linear equations

$\begin{array}{*{20}{c}} {x + 2y - 2z = 5} \\ {3x + by + z = 0} \\ { - x + y - 3z = a - 1.} \end{array}$
EXM.1.AHL.TZ0.6c:
Find D^–1.
EXM.2.AHL.TZ0.7a.ii:
Find $S$ ₁₀₀.
EXM.1.AHL.TZ0.35a:
Write down the inverse of the matrix

A = $\left( {\begin{array}{*{20}{c}} 1&{ - 3}&1 \\ 2&2&{ - 1} \\ 1&{ - 5}&3 \end{array}} \right)$
EXM.1.AHL.TZ0.10b:
det (2A − B).
EXM.1.AHL.TZ0.6b:
Find CD.
EXM.1.AHL.TZ0.49:
The square matrix X is such that X³ = 0. Show that the inverse of the matrix (I – X) is I + X + X².
EXM.1.AHL.TZ0.11c:
Find AB.
EXM.2.AHL.TZ0.21b.i:
find $x$ and $y$ in terms of $n$ .
EXM.1.AHL.TZ0.10a:
2A − B.
EXM.1.AHL.TZ0.4a:
Write down the inverse, A^–1.
20N.1.AHL.TZ0.F_4b.i:
Verify that

$A^{2} - α A + β I =$ 0.
EXM.1.AHL.TZ0.44:
Find the values of the real number $k$ for which the determinant of the matrix $\left( {\begin{array}{*{20}{c}} {k - 4}&3 \\ { - 2}&{k + 1} \end{array}} \right)$ is equal to zero.
EXM.1.AHL.TZ0.52b:
Hence solve the system of equations

$x + 2y + z = 0$

$x + y + 2z = 7$

$2x + y + z = 17$
EXM.2.AHL.TZ0.22b:
Hence show that M is a singular matrix.
EXM.2.AHL.TZ0.8d.ii:
Solve this matrix equation.
EXM.1.AHL.TZ0.22a:
Find AB.
EXM.1.AHL.TZ0.35c:
A fourth plane with equation $x + y + z = d$ passes through the point of intersection. Find the value of $d$ .
EXM.1.AHL.TZ0.52a:
Find the inverse of the matrix $\left( {\begin{array}{*{20}{c}} 1&2&1 \\ 1&1&2 \\ 2&1&4 \end{array}} \right)$ .
EXM.1.AHL.TZ0.30a:
Given that AB = $\left( {\begin{array}{*{20}{c}} a&0&0 \\ 0&a&0 \\ 0&0&a \end{array}} \right)$ , find $a$ .
EXM.1.AHL.TZ0.25a:
Find the matrix A².
EXM.2.AHL.TZ0.3b:
If ${M^2}$ is equal to $\left( {\begin{array}{*{20}{c}} 5&{ - 4} \\ { - 4}&5 \end{array}} \right)$ , find the value of $a$ .
EXM.1.AHL.TZ0.50:
Given that A = $\left( {\begin{array}{*{20}{c}} 2&3 \\ 1&{ - 2} \end{array}} \right)$ and B = $\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&{ - 3} \end{array}} \right)$ , find X if BX = A – AB.
EXM.2.AHL.TZ0.22d:
Show that (I − M)² = I − M where I is the identity matrix.
EXM.2.AHL.TZ0.23b.ii:
Hence show that I = $\frac{1}{5}$ A (6I – A).
EXM.1.AHL.TZ0.33:
Given that M = $\left( {\begin{array}{*{20}{c}} 2&{ - 1} \\ { - 3}&4 \end{array}} \right)$ and that M² – 6M + kI = 0 find k.
EXM.2.AHL.TZ0.22a.i:
Show that $a + d = 1$ .
EXM.2.AHL.TZ0.22a.ii:
Find an expression for $b c$ in terms of $a$ .
EXM.1.AHL.TZ0.48:
The matrices A, B, C and X are all non-singular 3 × 3 matrices.

Given that A^–1XB = C, express X in terms of the other matrices.
EXM.1.AHL.TZ0.11b:
Find −3A.
EXM.2.AHL.TZ0.10a.ii:
Find A².
EXM.2.AHL.TZ0.10d:
Let X be a 2 × 2 matrix such that AX = B. Find X.
20N.1.AHL.TZ0.F_4a:
By considering the determinant of a relevant matrix, show that the eigenvalues, $λ$ , of $A$ satisfy the equation

$λ^{2} - α λ + β = 0$ ,

where $α$ and $β$ are functions of $a, b, c, d$ to be determined.
18M.1.AHL.TZ0.F_2b:
Let B = $\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&{ - 3} \end{array}} \right]$ .

Given that B² – B – 4I = $\left[ {\begin{array}{*{20}{c}} k&0 \\ 0&k \end{array}} \right]$ , find the value of $k$ .
EXM.1.AHL.TZ0.2:
If $A = \left( {\begin{array}{*{20}{c}} {2p}&3 \\ { - 4p}&p \end{array}} \right)$ and det $A = 14$ , find the possible values of $p$ .
EXM.1.AHL.TZ0.29c:
Show that det(M²) is positive.
EXM.1.AHL.TZ0.32:
If A = $\left( {\begin{array}{*{20}{c}} 1&2 \\ k&{ - 1} \end{array}} \right)$ and A² is a matrix whose entries are all 0, find $k$ .
EXM.1.AHL.TZ0.26c:
Find A(A^–1B + 2A^–1)A.
EXM.1.AHL.TZ0.26a:
Find BA.
EXM.1.AHL.TZ0.25b:
If det A² = 16, determine the possible values of $a$ .

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