University Business Mathematics — Matrices and Input-Output Models Worksheet

All 13 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Modelling a Problem as a Linear System

   A bike shop assembles three models: city bikes, hybrids and cargo bikes. Each city bike needs $3$ hours of frame assembly, $1$ hour of wheel building and $1$ hour of testing; each hybrid needs $4$, $2$ and $1$ hours; each cargo bike needs $6$, $2$ and $3$ hours. Next week the shop has $120$ hours of frame assembly, $46$ hours of wheel building and $45$ hours of testing, and the owner wants every one of those hours used.

   1. Define your variables and write the system of equations, with the variables in the same order in every equation.
   2. The owner's first idea is $16$ city bikes, $12$ hybrids and $4$ cargo bikes; her assistant suggests $12$ city bikes, $9$ hybrids and $8$ cargo bikes. Check each plan against the system and say which one uses all the hours.
2. Q2Augmented Matrices and Row Operations

   Consider the system $$y+2x-z=5,3z-x=4,x+y+z=6.$$

   1. Write its augmented matrix, with the variables in the order $x,y,z$.
   2. Apply, in this order, $R_1\leftrightarrow R_3$, then $R_2\to R_2+R_1$, then $R_3\to R_3-2R_1$, and write the matrix after the three operations.
   3. Apply $R_3\to R_3+R_2$, then finish by back-substitution to find $x$, $y$ and $z$.
3. Q3Augmented Matrices and Row Operations

   For each pair, name the single row operation that turns the first augmented matrix into the second, written in the form $R_2\to R_2-3R_1$, $R_1\to \frac{1}{2}{R}_1$ or $R_1\leftrightarrow R_2$. If no single row operation does it, say so and explain why.

   1. $\left[\begin{array}{ccc}\hfill 1& \hfill 3& \hfill 7\\ \hfill 2& \hfill 5& \hfill 4\end{array}\right]⟶\backslash \left[\begin{array}{ccc}\hfill 1& \hfill 3& \hfill 7\\ \hfill 0& \hfill -1& \hfill -10\end{array}\right]$
   2. $\left[\begin{array}{ccc}\hfill 4& \hfill 8& \hfill 12\\ \hfill 1& \hfill 5& \hfill 2\end{array}\right]⟶\backslash \left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 1& \hfill 5& \hfill 2\end{array}\right]$
   3. $\left[\begin{array}{ccc}\hfill 0& \hfill 2& \hfill 6\\ \hfill 3& \hfill 1& \hfill 5\end{array}\right]⟶\backslash \left[\begin{array}{ccc}\hfill 3& \hfill 1& \hfill 5\\ \hfill 0& \hfill 2& \hfill 6\end{array}\right]$
   4. $\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 5\\ \hfill 3& \hfill 4& \hfill 6\end{array}\right]⟶\backslash \left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 5\\ \hfill 3& \hfill 4& \hfill 11\end{array}\right]$
4. Q4Gauss-Jordan Elimination for a Unique Solution

   Solve by Gauss-Jordan elimination, carrying the augmented matrix all the way to reduced row echelon form and naming each row operation: $$x+3y+z=10,2x+5y+4z=24,x+2y+4z=17.$$

5. Q5Gauss-Jordan Elimination for a Unique Solution

   A concert hall sold $1500$ tickets for one show in three sections: floor seats at $80, lower balcony at $60 and upper balcony at $35. Ticket revenue was $84 000, and the upper balcony sold twice as many tickets as the floor. Set up a system and solve it by Gauss-Jordan elimination to find how many tickets of each kind were sold.

6. Q6Systems with No Solution or Infinitely Many Solutions

   Solve each system by Gauss-Jordan elimination. Say whether it has no solution, exactly one solution or infinitely many solutions, and if there are infinitely many, write the solution set using a parameter $t$.

   1. $x+y+2z=5,3x+4y+5z=18,2x+3y+3z=10$
   2. $x-y+3z=2,2x-y+4z=7,x+z=5$
7. Q7Systems with No Solution or Infinitely Many Solutions

   An events company must staff a banquet with $20$ workers: chefs paid $30 an hour, servers paid $20 an hour and dishwashers paid $18 an hour. The client's budget fixes the total hourly wage bill at $440.

   1. Set up the system and solve it by Gauss-Jordan elimination, writing the general solution with the number of dishwashers $d$ as the parameter.
   2. Find the staffing if exactly $5$ dishwashers are hired.
   3. List every staffing that is possible in whole numbers of workers, with at least one worker of each kind.
8. Q8Matrix Sums, Scalar Multiples and Products

   A bike shop has two stores (rows: downtown, west end) and sells three models (columns: city, hybrid, cargo). For one month, $S$ is the stock at the start, $D$ the deliveries received and $T$ the bikes sold: $$S=\left[\begin{array}{ccc}12& 8& 5\\ 9& 6& 4\end{array}\right],D=\left[\begin{array}{ccc}10& 6& 4\\ 8& 10& 2\end{array}\right],T=\left[\begin{array}{ccc}15& 9& 6\\ 11& 12& 3\end{array}\right].$$

   1. Compute the end-of-month stock $E=S+D-T$ and say what the entry in row 2, column 3 means.
   2. Next month the owner will order $1.5$ times this month's deliveries. Compute $1.5D$.
9. Q9Matrix Sums, Scalar Multiples and Products

   A food truck sells poutine, sandwiches and smoothies at $12, $9 and $6; each costs it $5, $4 and $2 to make. Write these as the row matrices $P=\left[\begin{array}{ccc}12& 9& 6\end{array}\right]$ and $C=\left[\begin{array}{ccc}5& 4& 2\end{array}\right]$. Last weekend's sales were $$Q=\left[\begin{array}{cc}40& 55\\ 30& 20\\ 50& 70\end{array}\right]\text{(rows: poutine, sandwiches, smoothies; columns: Saturday, Sunday).}$$

   1. Compute $PQ$ and $CQ$, and say what each entry means.
   2. Compute $(P-C)Q$ and check that it equals $PQ-CQ$. What does it measure?
10. Q10The Inverse Matrix and Matrix Equations

    Let $A=\left[\begin{array}{ccc}1& 1& 0\\ 2& 3& 1\\ 1& 1& 1\end{array}\right]$.

    1. Find $A^{-1}$ by row-reducing the augmented block $[A\mid I_3]$ to $[I_3\mid A^{-1}]$, naming each row operation.
    2. Check your answer by computing $A{A}^{-1}$.
11. Q11The Inverse Matrix and Matrix Equations

    A sign shop makes two products: banners and vinyl decals. Each banner needs $2$ hours of design and $1$ hour of printing; each decal order needs $3$ hours of design and $2$ hours of printing. With $x$ banners and $y$ decal orders, the hours used are $AX$, where $$A=\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right].$$ For a $2\times 2$ matrix, if $ad-bc\ne 0$ then ${\left[}^{\begin{array}{cc}a& b\\ c& d\end{array}}={\displaystyle \frac{1}{ad-bc}}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$.

    1. Find $A^{-1}$.
    2. In week 1 the shop has $36$ design hours and $22$ printing hours; in week 2, $50$ and $30$. Use $A^{-1}$ to find the production that uses all the hours in each week.
12. Q12Leontief Input-Output Analysis

    A regional economy has two sectors, farming and food processing. Its technology matrix is $$M=\left[\begin{array}{cc}0.1& 0.4\\ 0.3& 0.2\end{array}\right]\text{(rows and columns: farming, food processing),}$$ where column $j$ lists the dollar value of each sector's output used up in producing $1 of sector $j$'s output. If $X$ is the total output and $D$ the final (outside) demand, both in millions of dollars, then $X=MX+D$, that is $(I-M)X=D$.

    1. Say in one sentence what the entry $0.3$ means.
    2. Find the total output of each sector needed to meet a final demand of $D=\left[\begin{array}{c}30\\ 60\end{array}\right]$.
13. Q13Synthesis — drawing on several topics in this unit

    A bakery makes batches of bread, croissants and cookies. The flour, butter and sugar (in kg) that one batch of each uses are the columns of $$A=\left[\begin{array}{ccc}3& 2& 2\\ 1& 2& 1\\ 1& 1& 1\end{array}\right]\text{(rows: flour, butter, sugar; columns: bread, croissants, cookies).}$$ Its stock is $B=\left[\begin{array}{c}64\\ 34\\ 26\end{array}\right]$ kg of flour, butter and sugar.

    1. The manager plans $10$ batches of bread, $8$ of croissants and $5$ of cookies. Compute the ingredients this uses and the stock left over.
    2. Find, by Gauss-Jordan elimination, the numbers of batches that would use the whole stock exactly.

The 8 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Modelling a problem as a linear system

Every question on this sheet that wears a business context starts the same way: decide what the unknowns are, give each a letter and a unit, and write one equation for each resource or condition the problem fixes. Q1 is that step on its own. Three bike models share three workshops, and every workshop's hours must be used exactly, so each workshop gives one equation: hours per bike of each model, times the number of bikes of that model, summed, equals the hours available.

Q1(b) then gives two candidate plans and asks which uses all the hours. Checking a plan means substituting it into every equation. A plan that satisfies two of the three is not a solution — it leaves some workshop idle or over-booked — so write out all three checks for both plans before you give a verdict.

Augmented matrices and the three row operations

An augmented matrix is the system with the letters stripped out: one row per equation, one column per variable, and a last column, after the bar, for the constants. Q2 starts from a system whose equations are written with the terms in scrambled order and with a variable missing from one of them, which is exactly where transcription goes wrong.

The three operations are the only moves allowed: swap two rows (R₁ ↔ R₃), multiply a row by a nonzero number (R₁ → ½R₁), and replace a row by itself plus a multiple of another row (R₂ → R₂ − 3R₁). None of them changes the solutions of the system. Q2(b) and (c) give you the operations in order and ask you to carry them out; apply each one to the matrix the previous one produced, not to the original, and finish with back-substitution from the last row up.

Q3 runs the idea backwards: given a before-and-after pair, name the operation. Compare the two matrices row by row. If one row is unchanged and the other changed, the changed row was either scaled or had a multiple of the unchanged row added to it — and a scaling multiplies every entry of the row by the same number, while an addition shifts each entry by a multiple of the matching entry in the other row. If both rows moved, look for a swap.

Gauss-Jordan elimination for a unique solution

Gauss-Jordan elimination is row reduction with a fixed plan, carried past echelon form until the coefficient part is the identity matrix. Then the last column simply reads off the solution.

Q4 is the method on a bare system, and it asks you to name each row operation. That record is worth the time: it is how a marker gives part marks, and it is how you find your own slip. Q5 puts the same method inside a ticket-sales problem. Three unknowns need three equations, and the problem supplies them in three different shapes — a total count, a total revenue, and a comparison between two sections.

Systems with no solution or infinitely many solutions

Row reduction does not always end at the identity. What the last rows look like tells you which of the three cases you are in, and Q6 asks you to decide for two systems that look equally ordinary before you start.

When there are infinitely many, the variable whose column has no pivot is free. Set it equal to the parameter t, then use each nonzero row to write a pivot variable in terms of t. The solution set is the list of all three variables as expressions in t, and substituting any value of t back into the original system is the check.

Q7 is the business version. Twenty workers and a fixed hourly wage bill give two equations for three kinds of worker, so the elimination cannot end with a unique answer, and part (a) names the free variable for you: the number of dishwashers, d. Part (b) is one substitution into the general solution. Part (c) is where the context takes over from the algebra.

Matrix sums, scalar multiples and products

Adding or subtracting matrices works entry by entry and needs matrices of the same size; a scalar multiple multiplies every entry. In Q8 the rows are stores and the columns are bike models in all three matrices, so an entry of S + D − T combines stock, deliveries and sales for one store and one model. Part (a) asks what one entry of the result means: say it with its row label, its column label and the time it refers to. Part (b) is the scalar multiple: every entry of D is multiplied by the same number, and the result keeps the rows and columns of D.

A product is defined only when the inside sizes match, and each entry is a row of the left matrix times a column of the right one: multiply matching entries and add. Q9 uses a one-row price matrix and a one-row cost matrix against a sales matrix whose rows are products and whose columns are days.

The inverse matrix and matrix equations

A square matrix A has an inverse A⁻¹ when AA⁻¹ = A⁻¹A = I. The way to find one here is row reduction: put A beside the identity and row-reduce the whole block until the left half is I.

Q10 asks for that on a 3 × 3 matrix, naming each operation, and then for the check AA⁻¹ = I. Every operation acts on all six columns of the block at once; the most common mistake is to reduce the left half and forget to apply the same move to the right half. If the left half ever produces a row of zeros, there is no inverse, and the method tells you so on its own.

Q11 shows why an inverse is worth having. The shop's hours used are AX, so a production plan that uses all the hours solves AX = B. Multiplying both sides on the left by A⁻¹ gives X = A⁻¹B. Part (a) is the 2 × 2 formula printed in the question: check that ad − bc is not zero, swap the diagonal entries, change the signs of the other two, and divide by ad − bc. Part (b) has two different weeks with the same A, and that is the point of the question — once A⁻¹ is known, each week is one matrix product rather than a new elimination.

Leontief input-output analysis

An economy's sectors use one another's output to produce their own. The technology matrix M records that: column j lists how many dollars of each sector's output are used up in making one dollar of sector j's output. So a column belongs to the sector doing the consuming, and a row to the sector doing the supplying.

Total output X has to cover two uses: what the sectors consume from each other, MX, and the final demand D from outside. That balance is X = MX + D, which rearranges to (I − M)X = D. Q12(b) is then a linear system. Form I − M by subtracting each entry of M from the identity — the diagonal becomes one minus the entry, the off-diagonal entries change sign — and solve, either by the 2 × 2 inverse formula from Q11 or by Gauss-Jordan elimination.

The synthesis question

Q13 uses both halves of the set on one bakery. The columns of A are recipes, so A times a column of batch counts is a column of ingredients used. Part (a) is that product for the manager's plan, followed by a subtraction from the stock B, and each entry of the leftover column should be stated in kilograms of a named ingredient. Part (b) asks for the batches that use the stock exactly, which is the system AX = B, solved by Gauss-Jordan elimination on the augmented matrix [A | B]. Before you trust the answer, ask whether it makes sense as batches of baking, and check it by multiplying A by it.

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Getting the most out of it

Build the table before the equations

For every word problem, draw a small grid — resources down the side, products across the top — and fill it from the text. The equations, the augmented matrix and the matrix product all come straight off that grid, in the same order.

Name every row operation

Write R₂ → R₂ − 2R₁ beside the matrix it produced. It costs seconds, it is what earns part marks, and it lets you find the one line where an arithmetic slip crept in instead of starting over.

Check by substitution or by multiplication

A solution goes back into the original equations; an inverse is multiplied by the matrix it came from. Both checks are faster than the work they verify, and an elimination answer is never finished without one.

Say what every entry means

Whenever a matrix is the answer, pick one entry and say what it measures, with units and labels. If you cannot, the product was probably taken in the wrong order or the matrices were set up with rows and columns swapped.

Common questions