Competency-based application questions

MATExam Practice (Competency)

Competency-based questions translate real-world data (sales, costs, production, ticket pricing) into matrices and extract answers through matrix multiplication or the matrix-inverse method. The core idea is to arrange quantities so the inner dimensions match: a $(m \times n)$ data matrix times an $(n \times 1)$ price/cost column gives an $(m \times 1)$ result of totals per row.

Boards favour these because they test whether you can MODEL a situation, not just compute, and they reward correct order of multiplication and clear labelling of rows/columns.

Total per row (revenue / cost)

\begin{pmatrix} q_{11} & q_{12} & q_{13} \\ q_{21} & q_{22} & q_{23} \end{pmatrix} \begin{pmatrix} p_1 \\ p_2 \\ p_3 \end{pmatrix} = \begin{pmatrix} T_1 \\ T_2 \end{pmatrix}

q_{ij}

is quantity of product

j

for entity

i

;

p_j

is price/cost of product

j

;

T_i

is the total for entity

i

(e.g. each store or factory).

Conformability for multiplication

A_{m \times n}\, B_{n \times p} = C_{m \times p}

Product

AB

exists only when columns of

A

equal rows of

B

; the result has order rows-of-

A

by columns-of-

B

A

is the quantity matrix,

B

the price column.

Linear system as matrix equation

AX = B \quad \Rightarrow \quad X = A^{-1}B

A

is the coefficient matrix,

X

the column of unknown prices,

B

the column of given totals; valid only when

|A| \ne 0

Inverse of a 2x2 matrix

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow A^{-1} = \dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Used to solve

AX=B

; requires

ad-bc \ne 0

. Swap the diagonal entries, negate the off-diagonal, divide by the determinant.

Decide row vs column meaning FIRST: rows usually = entities (stores, factories, meals), columns = products; the price/cost vector is then a column $(n \times 1)$ .
For totals per entity, multiply quantity matrix (on the left) by the price column (on the right): order $A_{m\times n}B_{n\times 1}=C_{m\times 1}$ .
Always check conformability before multiplying; if dimensions clash, you have placed prices as a row instead of a column or vice-versa.
Each entry of the answer is a sum of products $\sum_j q_{ij}p_j$ , i.e. one dot product of a quantity row with the price column.
For 'find the prices' problems, set up $AX=B$ from the worded equations and solve by $X=A^{-1}B$ when $|A|\ne 0$ .
Compare results (cheaper meal, higher revenue store) only AFTER computing; state the comparison explicitly as the final sentence — competency marks require interpretation.
Keep units consistent (all in Rs) and label the final column vector entries with the entity names.

Where the marks go

Writing the price vector as a row and the quantity matrix on the right, getting a non-conformable or wrongly-ordered product.
Computing $BA$ instead of $AB$ — since matrix multiplication is not commutative, the wrong order gives a meaningless or differently-sized matrix.
Forgetting to interpret the numerical answer (which store/meal) — boards deduct marks for an un-concluded competency answer.
Using the inverse method when $|A|=0$ ; a singular coefficient matrix means no unique price solution, which must be stated rather than forced.

How the board asks it

Applicationtotals per row from a data matrix times a price column
A bookshop chain sells $3$ types of notebooks. Store $P$ sold $40$ , $25$ , $30$ units and store $Q$ sold $35$ , $40$ , $20$ units, priced at $\text{Rs }50$ , $\text{Rs }80$ and $\text{Rs }120$ respectively. Using matrix multiplication, find the total revenue of each store and state which store earned more.
Numericalmatrix-inverse method $X=A^{-1}B$ when $|A|\ne 0$
The cost of $2$ pens and $3$ pencils is $\text{Rs }30$ , while $3$ pens and $2$ pencils cost $\text{Rs }35$ . Express the situation as a matrix equation $AX=B$ and solve it by $X=A^{-1}B$ to find the price of one pen and one pencil.
Give reasonsconformability for multiplication (inner dimensions must match)
A student writes the price list as a row matrix of order $(1\times 3)$ and the quantity data as a matrix of order $(2\times 3)$ , then tries to multiply them. Give reasons why this product is not defined and state the correct order in which the two matrices must be multiplied to obtain the totals.
Assertion–Reasonsingular coefficient matrix, $|A|=0$
Assertion: The system of equations modelling the prices of two meals has no unique solution. Reason: The coefficient matrix $A$ satisfies $|A|=0$ , so $A^{-1}$ does not exist. State whether both statements are true and whether the Reason correctly explains the Assertion.
Applicationchoosing row vs column meaning before multiplying
A factory's daily output of $2$ products on $3$ days is to be combined with the per-unit profit of each product to find the total profit per day. Set up the output data and the profit list as matrices of the correct orders, write the product that gives a $(3\times 1)$ column of daily profits, and label each entry with its day.

PreviousInverse of a matrix by elementary operations

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.