Matrices (DP IB Applications & Interpretation (AI): HL): Exam Questions

Consider the following matrices:

$\mathit{A}=\left(\begin{array}{cc}x& 7\\ -4& 3\end{array}\right)\mathit{B}=\left(\begin{array}{cc}2& y\\ 4& 8\end{array}\right)$

$\mathit{C}=\left(\begin{array}{cc}1& a\\ 4& 8\end{array}\right)\mathit{D}=\left(\begin{array}{cc}b& 2\\ -9& -10\end{array}\right)\mathit{E}=\left(\begin{array}{cc}-34& 62\\ -66& -52\end{array}\right)$

$\mathit{M}=\left(\begin{array}{cc}3& 2c\\ -c& 6\end{array}\right)\mathit{N}=\left(\begin{array}{cc}10& d+24\\ d& 22\end{array}\right)$

where $a,b,c,d,x,y\in \mathrm{ℝ}$ are constants.

Given that $\mathit{A}+q\mathit{B}=z\mathit{I}$,  find the values of x, y, z, and q.

Given that $\frac{1}{2}\mathit{E}=r\mathit{C}+s\mathit{D}$   find the values of a, b, r, and s.

Given that $e\mathit{M}-f\mathit{N}=\mathit{I}$ find the values of c, d, e, and f.

Consider the two matrices

 $\mathit{A}=\left(\begin{array}{cc}3& 2\\ 2& 0\end{array}\right)$

$B=\left(\begin{array}{cc}13& 6\\ 6& q\end{array}\right)$

where $q\in \mathrm{ℝ}$ is a constant. 

Given that $\mathit{A}$and $\mathit{B}$ are commutative, find the value of $q$.

Consider the matrix

$\mathit{M}=\left(\begin{array}{cc}3& 0\\ 0& -2\end{array}\right)$ 

Find an expression for ${\mathit{M}}^k$ .

Consider the matrices:

 $\mathit{A}=\left(\begin{array}{cc}2& a\\ 3& 4\end{array}\right)\mathit{B}=\left(\begin{array}{ccc}1& 3& 4\\ 3& -2& b\end{array}\right)\mathit{C}=\left(\begin{array}{ccc}1& 2& 3\\ 4& c& 5\\ -5& -3& -2\end{array}\right)$

$\mathit{D}=\left(\begin{array}{cc}d& 1\\ 2& 3\end{array}\right)\mathit{E}=\left(\begin{array}{ccc}3& 2& -1\\ e& -2& 4\\ 1& -3& 2\end{array}\right)\mathit{F}=\left(\begin{array}{cc}2& f\\ -2& -1\\ 3& 2\end{array}\right)$

where $a,b,c,de,f\in \mathrm{ℝ}$ are constants. 

Find the following products in terms of the appropriate constants.  If it is not possible to do so, explain why.

(i) $\mathit{A}\mathit{B}$

(ii) $\mathit{C}\mathit{A}$

(iii) $\mathit{E}\mathit{B}$

(iv) $\mathit{B}\mathit{E}$

(v) $\mathit{E}\mathit{A}\mathbf{+}\mathit{F}$

(vi) $\mathit{F}\left(C-E\right)$

(vii) $\left(\mathit{C}\mathbf{-}\mathit{E}\right)\mathit{F}$

Consider the following matrices:

 $\mathit{M}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\mathit{N}=\left(\begin{array}{cc}e& f\\ g& h\end{array}\right)\mathit{P}=\left(\begin{array}{cc}i& j\\ k& l\end{array}\right)$ 

Show that $\left(\mathit{M}\mathit{N}\right)\mathit{P}=\mathit{M}\left(\mathit{N}\mathit{P}\right)$ and state the name of this property.

For each of the following matrices, 

(i)     find the values of $x$ for which ${\mathit{M}}^{-1}$ does not exist, and 

(ii)    for the cases where ${\mathit{M}}^{-1}$ does exist, find ${\mathit{M}}^{-1}$ in terms of $x$.

$\mathit{M}=\left(\begin{array}{cc}x& -2\\ 1& 3\end{array}\right)$

$\mathit{M}=\left(\begin{array}{cc}x-2& 3\\ 4& 2x\end{array}\right)$

$\mathit{M}=\left(\begin{array}{cc}{x}^2& x-1\\ 3x& 5\end{array}\right)$

A message is encoded using a matrix. Letters in the message are represented by numbers as given in the table below.

Messages are encoded by splitting the message into pairs of letters and writing in a $2\times n$  matrix.  For example, “encode” becomes $\left(\begin{array}{ccc}\mathrm{E}& \mathrm{C}& \mathrm{D}\\ \mathrm{N}& \mathrm{O}& \mathrm{E}\end{array}\right)$ or $\left(\begin{array}{ccc}5& 3& 4\\ 14& 15& 5\end{array}\right)$. Then the message is multiplied, on the left, by an encryption matrix.

Here is a message which has been encoded using the encryption matrix $\left(\begin{array}{cc}3& 2\\ -1& 4\end{array}\right)$:

$\left(\begin{array}{cccccc}41& 96& 33& 53& 96& 46\\ -9& 52& 3& 71& 66& 8\end{array}\right)$ 

Decode the message.

A spy (whose name cannot be given) notices that if the message has an odd number of letters, it will not completely fill a $2\times n$ matrix. The spy suggests putting the message into a $3\times n$ matrix if the number of letters in the message is a multiple of three.

Assuming the same encryption matrix is used as was used for the above message, explain the problem with the spy’s suggestion.

Using the properties of matrices, explain why the following misconceptions are incorrect.

“I know that $\left(a+b\right)\left(a-b\right)=a^2-b^2$,  therefore  $\left(\mathit{A}+\mathit{B}\right)\left(\mathit{A}-\mathit{B}\right)={\mathit{A}}^{\mathbf{2}}-{\mathit{B}}^{\mathbf{2}}$ must also be true if $A$ and $B$ are matrices.”

“I know that ${\left(a+b\right)}^2=a^2+2ab+b^2$,  therefore ${\left(\mathit{A}+\mathit{B}\right)}^2={\mathit{A}}^{\mathbf{2}}+\mathbf{2}\mathit{A}\mathit{B}+{\mathit{B}}^{\mathbf{2}}$ must also be true if $A$  and $B$ are matrices.”

In this question $\mathit{A}$ and $\mathit{B}$ are $2\times 2$  matrices, $k,p,q\in \mathrm{ℝ}$ are constants, and $n$ is a positive integer.

(i) Show that $\left(p\mathit{A}\right)\left(q\mathit{B}\right)=pq\left(\mathit{A}\mathit{B}\right)$.

(ii) Hence show that ${\left(k\mathit{A}\right)}^n=k^n{\mathit{A}}^n$. 

A student claims it is always true that $\left(\mathit{A}\mathit{B}\right)\text{'}\text{'}=\mathit{A}\text{'}\text{'}\mathit{B}\text{'}\text{'}$.

Either explain why the student’s claim is always true, or else show that it is not always true by providing a counterexample.

Consider the following matrices:

 $\mathit{A}=\left(\begin{array}{cc}5& p\\ p& 7\end{array}\right)\mathit{B}=\left(\begin{array}{cc}11& q\\ q& 3\end{array}\right)$ 

where $p,q\in \mathrm{ℝ}$ are constants, $q>0.$

Given that $\mathit{A}$and $\mathit{B}$ are commutative, find $q$ in terms of $p.$

Given that the determinant of $\mathit{A}$is 26, find $p$ and $q$.

A couple are planning their wedding reception and wish to buy a bow for every chair, with a matching tablecloth for each table. 

There are a total of 120 chairs and 15 tables, all of which need bows and tablecloths respectively. 

Company A charges £1.03 per chair bow and £14 per tablecloth.

Company B charges £0.85 per chair bow and £16 per tablecloth. 

By setting up one matrix equation that includes both companies, compare the overall prices that would be charged by the two companies.

Married friends of the couple recommend Company C, whom they used for their wedding. The friends can remember that they paid £13 per tablecloth, but cannot remember the price per chair bow.

Set up and solve a matrix equation to find the maximum price per chair bow that Company C could charge so as still to be cheaper overall than companies A and B.

Veterinarians, veterinary nurses, and animal care assistants are paid fixed salaries according to an industry standard. The totals of the annual payrolls for three veterinary practices that pay according to the industry standard are summarised in the table below.

Using matrices, set up and solve a system of equations to find the fixed salaries that are paid for each of the three roles.

Vicky is setting up a new veterinary practice, and to help recruit staff she is planning to pay 5% above the industry standard for all job roles.  She uses the following matrix multiplication to help find the total cost of her staffing, where $p,q$ and $r$ represent the salaries for a veterinarian, a veterinary nurse, and an animal care assistant respectively:

 $a\times \left(\begin{array}{ccc}3& 4& 2\end{array}\right)\times \left(\begin{array}{c}p\\ q\\ r\end{array}\right)$= Total salary spend in thousands

(i) Write down the value of the constant $a$ that Vicky should use

(ii) Interpret the meaning of the element ‘4’ in the row matrix.

Consider a curve with equation $y=a{x}^3+b{x}^2+cx-8$, where a, b, and c are real constants.  The graph passes through the points $\mathrm{A}\left(2,116\right),\mathrm{B}\left(-4,-712\right)$ and $\mathrm{C}\left(3,394\right).$ 

(i) Use a matrix method to find the values of $a,b$and $c$.

(ii) Hence sketch the curve.

Consider a second curve with equation $a{x}^5+b{x}^4+c{x}^3+d{x}^2+ex+f$,  where a, b, c, d, e and f are real constants with $a\ne 0.$ 

By considering the method used to solve part (a), suggest the number of coordinates that would need to be known to determine the values of all the constants in the equation of the second curve.

Consider the following matrices:

 $\mathit{A}=\left(\begin{array}{ccc}a& 4& 5\\ 1& 2& 6b\end{array}\right)\mathit{B}=\left(\begin{array}{cc}1& 3\end{array}\right)\mathit{C}=\left(\begin{array}{c}-2\\ -1\\ 0\end{array}\right)$ 

The matrix product $\left(\mathit{M}\mathit{N}\right)\mathit{P}$ is calculated, where M, N, and P can each be any one of the matrices A, B or C.

Find the possible dimensions of the resulting products.

The matrix sum M + N + P   is calculated, where M, N, and P can each be any one of the matrices A, B or C.

Find the possible sums that could result from such an addition.

It is given that $\left(\mathit{C}\mathit{B}\right)\mathit{A}=t\left(\begin{array}{ccc}2& 4& 9.2\\ 1& 2& 4.6\\ 0& 0& 0\end{array}\right).$

Find the values of a, b, and t.

In this question A, B, and C are arbitrary square matrices.

Prove the following matrix results, stating any necessary assumptions:

(i) If  $\mathit{A}\mathit{B}=\mathit{C}$ then $\mathit{B}={\mathit{A}}^{-1}\mathit{C}$ 

(ii) ${\left(\mathit{A}\mathit{B}\right)}^{-1}={\mathit{B}}^{-1}{\mathit{A}}^{-1}$ 

Using the result from part (a)(ii), simplify ${\left({\mathit{A}}^{-1}\mathit{B}\right)}^{-1}{\mathit{A}}^{-1}$.

Consider the matrices:

 $\mathit{M}=\left(\begin{array}{cc}3& 2\\ 2& 0\end{array}\right)$

$\mathit{N}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ 

where $a,b,c,d\in \mathrm{ℝ}$ are constants. 

Given that M and N are commutative, find an expression for N in terms of b and d only.

Given that the inverse matrix ${\left(\mathit{M}\mathit{N}\right)}^{-1}$ exists 

(i) determine a relationship that the constants $a,b,c$and $d$ must satisfy 

(ii) find ${\left(\mathit{M}\mathit{N}\right)}^{-1}$ in terms of $a,b,c$ and $d$.

Given that

 $\mathit{A}=\left(\begin{array}{cc}2& 8\\ 2& -4\end{array}\right)$

${\mathit{A}}^{-1}\mathit{B}\mathit{A}=\left(\begin{array}{cc}2& 0\\ 5& 6\end{array}\right)$

$\mathit{A}\mathit{B}\mathit{C}=3\mathit{I}$

find matrices B and C.

For any $2\times 2$ matrices M, A or B:

Prove that $\det \left(k\mathit{M}\right)=k^2\det \mathit{M}$,  where $k$ is a real constant.

Prove that $\det \left(\mathit{A}\mathit{B}\right)=\det \mathit{A}\times \det \mathit{B}$.

Consider the matrix

 $\mathit{M}=\left(\begin{array}{cc}a& 0\\ c& -a\end{array}\right)$ 

where $a,c\in \mathrm{ℝ}$ . Find expressions for ${\mathit{M}}^{2k}$ and ${\mathit{M}}^{2k+1}$ where k is a positive integer.

Consider the matrix

 $\mathit{A}=\left(\begin{array}{cc}i& 0\\ 3& i\end{array}\right)$ 

where $i=\sqrt{-1}$.

(i) Find ${\mathit{A}}^2,{\mathit{A}}^3,{\mathit{A}}^4$ and ${\mathit{A}}^5$.

(ii) Hence determine ${\mathit{A}}^{15}$.

Find the general term for $\mathit{A}\text{'}\text{'}$where $n$ is a positive integer.

Consider the $2\times 2$ matrix

 $\mathit{M}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ 

Use algebra to find the requirements that must be satisfied by $a,b,c$ and $d$ in order for ${\mathit{M}}^2=\left(\begin{array}{cc}{a}^2& b^2\\ c^2& d^2\end{array}\right)$ to be true.

A professional Football team are looking to buy new players.  Their scouts have returned a shortlist containing 23 English, 17 German, 18 Spanish, and 8 Italian players.

The shortlisted players are in the following proportions for each playing position: 

• 11% are goalkeepers

• 29% are wingers

• 39% are defenders

• 21% are strikers 

A given player only plays in one of the listed positions. 

(i) Write a column matrix, N, representing the numbers of players from each country, and a row matrix, P, containing the proportions of players in each position.

(ii) Hence find the total number of players in each position on the shortlist.

Explain why it would be incorrect in general to say that the elements of the matrix NP  represent the numbers of players in each position by nationality. For example to say that (NP)_1,1 (the entry in the first row and first column of matrix NP ) might represent the number of English goalkeepers.

A ball is thrown vertically downwards from the top of a cliff, and its position is tracked from when it is first thrown until it hits the ground (at which point it may be assumed that the ball comes instantaneously to rest).

The height of the ball above the ground after $t$ seconds is modelled by the equation $s\left(t\right)=a{t}^2+bt+c$,  where a, b and c are real constants and the height s is measured in metres. After 1 second, the ball is 175.4 m above the ground; after 5 seconds, its height is 105 m; and after 6 seconds it is 74.4 m. 

Set up and solve a matrix equation to find

(i) the height of the cliff

(ii) the time taken for the ball to reach the ground.

Another experiment studies the motion of another object that only moves in one dimension.  A quartic equation of the form $s=a{t}^4+b{t}^3+c{t}^2+dt+e$ is used to model the displacement of the object, where $a,b,c,d$ and $e$ are all real constants with $a\ne 0.$ 

(i) State the number of measurements of the object’s displacement at different times that would need to be taken, in order to find the explicit values of the constants $a,b,c,d$ and $e$.

(ii) State the dimensions of the matrices involved in forming a matrix equation, as in part (a), to determine the constants for the quartic model.