Worked examples — Matrix definition — rows, columns, order, elements

This page is a drill hall. The parent note taught you the four ideas — rows, columns, order, elements. Here we hit every kind of question those ideas can produce, one worked example per scenario, so you never meet a matrix problem you have not already seen.

Before we start, one promise about notation. Everything we use is built here:

• A matrix is a grid of numbers inside brackets.
• The order $m\times n$ means "$m$ rows tall, $n$ columns wide" — read rows first.
• $a_{ij}$ means "the number sitting in row $i$, column $j$" — again row first, column second ("RC").
• $i$ and $j$ always start at 1, never at 0.

If any of those feel shaky, the parent note builds them from the ground up. From here on we use them at speed.

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The scenario matrix

Every matrix-reading or matrix-building question you can be asked falls into one of these cells. The worked examples below are labelled with the cell they cover, so together they fill the whole table.

Cell	Scenario	Covered by
A	Read an element from a small square grid	Example 1
B	Rectangular (non-square) grid — order not commutative	Example 2
C	Degenerate shapes: a single number, a row vector, a column vector	Example 3
D	The empty / edge index — asking for an element that does not exist	Example 4
E	Build a matrix from a formula $a_{ij}=f(i,j)$ with signs & a $\max /\min$ twist	Example 5
F	Sign-sensitive build: alternating $(-1{)}^{i+j}$ pattern (every sign case)	Example 6
G	Real-world word problem — turn a table of data into a matrix and read it	Example 7
H	Exam twist — two matrices equal, solve for unknowns (links to equality)	Example 8

Related ideas we brush against: Matrix Equality, Types of Matrices, Transpose of Matrix.

Look at the figure above: the same $2\times 3$ grid with the red arrow going down first (that picks the row $i$) and the blue arrow going across (that picks the column $j$). That single picture is the whole grammar of $a_{ij}$. Keep it in your mind's eye for every example.

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[!example] Example 1 — Cell A: read a square grid

$P=\left[\begin{array}{ccc}8& -3& 5\\ 0& 4& -6\\ 2& 9& -1\end{array}\right]$

Find the order, then $p_{13}$, $p_{31}$ and the diagonal element $p_{22}$.

Step 1 — Count rows, then columns. 3 horizontal lines, 3 vertical lines → order $3\times 3$. Why this step? Shape comes before contents; you cannot ask for row 4 in a 3-row matrix.

Step 2 — $p_{13}$: go down to row 1, across to column 3. Row 1 is $[8-35]$; the 3rd entry is $5$. So $p_{13}=5$. Why this step? Red arrow (down) picks the row, blue arrow (across) picks the column — exactly the figure.

Step 3 — $p_{31}$: go down to row 3, across to column 1. Row 3 is $[29-1]$; the 1st entry is $2$. So $p_{31}=2$. Why this step? Swapping the subscripts moves you to a completely different cell — $5\ne 2$, so $p_{13}\ne p_{31}$.

Step 4 — $p_{22}$: row 2, column 2. Row 2 is $[04-6]$; 2nd entry is $4$. So $p_{22}=4$.

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[!example] Example 2 — Cell B: rectangle, order is not commutative

$Q=\left[\begin{array}{cccc}5& -3& 0& 1\\ 2& 7& -4& 6\end{array}\right]$

Find the order, the total number of elements, $q_{14}$, $q_{23}$, and state whether $Q$ is square.

Step 1 — Order. 2 rows, 4 columns → order $2\times 4$. Why this step? $2\times 4$ and $4\times 2$ are different shapes; unlike ordinary multiplication where $2\times 4=4\times 2$, matrix order does not commute.

Step 2 — Total elements $=m\times n=2\times 4=8$. Why this step? Each of the 2 rows holds 4 numbers, and $2\times 4$ counts them without overlap.

Step 3 — $q_{14}$: row 1, column 4. Row 1 is $[5-301]$; 4th entry is $1$. So $q_{14}=1$.

Step 4 — $q_{23}$: row 2, column 3. Row 2 is $[27-46]$; 3rd entry is $-4$. So $q_{23}=-4$.

Step 5 — Square? Square needs rows $=$ columns. Here $2\ne 4$, so not square (it is rectangular).

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[!example] Example 3 — Cell C: the degenerate shapes

The smallest matrices strip the idea down to its bones. Three degenerate cases:

(a) A single number as a matrix $S=\left[\begin{array}{c}7\end{array}\right]$. Order $1\times 1$. Only element $s_{11}=7$. This is a square matrix (rows $=$ columns $=1$).

(b) A row vector $R=\left[\begin{array}{cccc}3& -1& 5& 0\end{array}\right]$. Order $1\times 4$: one row, four columns. Here the first index is stuck at 1, only the second moves: $r_{11}=3,r_{12}=-1,r_{13}=5,r_{14}=0$.

(c) A column vector $C=\left[\begin{array}{c}2\\ -7\\ 4\end{array}\right]$. Order $3\times 1$: three rows, one column. Now the second index is stuck at 1: $c_{11}=2,c_{21}=-7,c_{31}=4$.

Why care about these? They are the limiting cases of the whole topic: a row vector is "a matrix that ran out of rows", a column vector "ran out of columns", and a $1\times 1$ is "both at once". Every general rule must still work here, and it does — the frozen index just never changes.

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[!example] Example 4 — Cell D: the element that does not exist

Using $Q$ from Example 2 (order $2\times 4$), a question asks for $q_{34}$ and for $q_{25}$. Find them.

Step 1 — Test $q_{34}$: row 3. Order is $2\times 4$, so valid rows are $i\in \{1,2\}$. Row $3$ is out of range. Why this step? The index $i$ can never exceed $m$; asking for row 3 in a 2-row matrix has no answer — the element is undefined, not zero.

Step 2 — Test $q_{25}$: column 5. Valid columns are $j\in \{1,2,3,4\}$. Column $5$ exceeds $n=4$. Why this step? The second index can never exceed $n$. $q_{25}$ is undefined.

Step 3 — State the allowed ranges. For any matrix of order $m\times n$: $1\le i\le m$ and $1\le j\le n$. Anything outside is a non-existent element.

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[!example] Example 5 — Cell E: build from a formula with a twist

Construct the $3\times 3$ matrix $A$ with $a_{ij}=\max (i,j)-\min (i,j).$ (The term $\max (i,j)$ means "the bigger of $i$ and $j$"; $\min (i,j)$ the smaller. Their difference is just how far apart the two indices are.)

Step 1 — Diagonal, $i=j$. $\max (i,i)-\min (i,i)=i-i=0$. So $a_{11}=a_{22}=a_{33}=0$. Why this step? When the row and column indices agree, the gap is zero — a whole diagonal of $0$s for free.

Step 2 — Off-diagonal. The value is $|i-j|$, the distance between the two indices:

• $a_{12}=|1-2|=1$, $a_{13}=|1-3|=2$, $a_{23}=|2-3|=1$
• $a_{21}=|2-1|=1$, $a_{31}=|3-1|=2$, $a_{32}=|3-2|=1$

Why this step? $\max -\min$ of two numbers is exactly their absolute difference, which cannot be negative — so this build has no sign traps.

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

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[!example] Example 6 — Cell F: every sign, alternating pattern

Construct the $3\times 3$ matrix $B$ with $b_{ij}=(-1{)}^{i+j}(i+j).$ Here $(-1{)}^{i+j}$ is a sign switch: it equals $+1$ when $i+j$ is even and $-1$ when $i+j$ is odd. We must handle both sign cases correctly.

Step 1 — Read the sign rule. $i+j$ even $\Rightarrow$ factor $+1$; $i+j$ odd $\Rightarrow$ factor $-1$. Why this step? Getting the parity right in each cell is the whole difficulty; the magnitude $(i+j)$ is easy.

Step 2 — Row 1 ($i=1$):

• $b_{11}$: $i+j=2$ even → $+(2)=2$
• $b_{12}$: $i+j=3$ odd → $-(3)=-3$
• $b_{13}$: $i+j=4$ even → $+(4)=4$

Step 3 — Row 2 ($i=2$):

• $b_{21}$: $3$ odd → $-3$
• $b_{22}$: $4$ even → $4$
• $b_{23}$: $5$ odd → $-5$

Step 4 — Row 3 ($i=3$):

• $b_{31}$: $4$ even → $4$
• $b_{32}$: $5$ odd → $-5$
• $b_{33}$: $6$ even → $6$

Step 5 — Assemble. $B=\left[\begin{array}{ccc}2& -3& 4\\ -3& 4& -5\\ 4& -5& 6\end{array}\right]$

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[!example] Example 7 — Cell G: a real-world word problem

A snack stall records sales over 3 days for 2 items (samosas, chai):

• Monday: 40 samosas, 55 chai
• Tuesday: 30 samosas, 48 chai
• Wednesday: 52 samosas, 60 chai

Write this as a matrix $D$ with rows = days and columns = items. Then read $d_{31}$ and $d_{12}$, and say what each means.

Step 1 — Fix the layout. Rows = days (3 of them), columns = items (2 of them) → order $3\times 2$. Why this step? Choosing what rows and columns mean is the modelling step; every element then has a definite real-world reading.

Step 2 — Fill the grid (samosas in column 1, chai in column 2): $D=\left[\begin{array}{cc}40& 55\\ 30& 48\\ 52& 60\end{array}\right]$

Step 3 — Read $d_{31}$: row 3 (Wednesday), column 1 (samosas) $=52$. Meaning: 52 samosas sold on Wednesday.

Step 4 — Read $d_{12}$: row 1 (Monday), column 2 (chai) $=55$. Meaning: 55 chai sold on Monday.

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[!example] Example 8 — Cell H: exam twist, solve for unknowns

Two matrices are given as equal (they are equal matrices, meaning same order and every matching entry equal): $\left[\begin{array}{cc}x+y& 3\\ 5& x-y\end{array}\right]=\left[\begin{array}{cc}8& 3\\ 5& 2\end{array}\right].$ Find $x$ and $y$.

Step 1 — Match entry by entry. Equal matrices force each corresponding element equal:

• top-left: $x+y=8$
• top-right: $3=3$ (automatically true — no info)
• bottom-left: $5=5$ (automatically true)
• bottom-right: $x-y=2$

Why this step? Matrix equality is entry-wise; the two useful cells give a pair of ordinary equations.

Step 2 — Solve the system. Add the two equations: $(x+y)+(x-y)=8+2\Rightarrow 2x=10\Rightarrow x=5.$ Why this step? Adding cancels $y$, isolating $x$ immediately.

Step 3 — Back-substitute. From $x+y=8$: $5+y=8\Rightarrow y=3$.

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[!mnemonic] Carry these away

• RC: Row before Column, always. Down first, then across (see the two arrows in the figure).
• Order never commutes: $m\times n\ne n\times m$ as a shape, even though the count $m\cdot n$ is the same.
• Indices live in a box: $1\le i\le m$, $1\le j\le n$. Outside the box → the element does not exist.
• Build formulas: write the blank grid first, then feed each $(i,j)$ in — and watch parity for any $(-1{)}^{i+j}$.

[!recall]- Quick self-test