Scalar multiplication: Worked examples — Maths deep dive

This is the "throw everything at it" page for scalar multiplication. The parent note built the rule: $k A$ means multiply every entry of $A$ by the number $k$ . Here we drill that rule against every kind of input it can meet — positive, negative, zero, fractions, the zero matrix, a word problem, and an exam trap.

Before we start, one word we lean on: a scalar is just a single ordinary number (like $3$ , $- 2$ , or $\frac{1}{2}$ ) — see vectors and scalars. It is not a matrix. Every example below multiplies a matrix (a grid of numbers, introduction to matrices) by one such scalar.

The scenario matrix

Every scalar-multiplication problem you can be handed falls into one of these boxes. We will hit all of them.

Cell	Scenario	What's tricky about it	Example
A	Positive whole scalar, $k > 1$	Baseline — stretch every entry	Ex 1
B	Fractional scalar, $0 < k < 1$	Shrink; watch fractions	Ex 2
C	Negative scalar, $k < 0$	Every sign flips	Ex 3
D	The scalar $k = 0$	Collapses to zero matrix $O$	Ex 4
E	The scalar $k = 1$	Nothing changes (identity check)	Ex 4
F	Combined: $k_{1} A + k_{2} B$	Scale twice, then add	Ex 5
G	Geometric meaning (columns as arrows)	Scaling = stretch/reflect the picture	Ex 6
H	Real-world word problem (units!)	Translate words → matrix → scale	Ex 7
I	Exam twist: solve for $k$ or $X$	Work the rule backwards	Ex 8

Let's go cell by cell.

Cell A — Positive whole scalar

Cell B — Fractional scalar (shrink)

Cell C — Negative scalar (every sign flips)

Cells D & E — the two special scalars $0$ and $1$

Cell F — combined scaling then addition

Cell G — the geometric picture

Here the two columns of a $2 \times 2$ matrix are read as arrows starting at the origin. Scaling the matrix scales both arrows. This is the seed of linear transformations.

Worked example Example 6 — stretch and reflect arrows (Cell G)

Let $M = [21 - 1 1]$ . Its columns are the arrows $u = [21]$ and $v = [- 1 1]$ . Compute $2 M$ and $(- 1) M$ and describe the pictures.

Forecast: doubling should make each arrow twice as long, same direction. Multiplying by $- 1$ should flip each arrow to point the opposite way. Picture it before the figure.

Scale by $2$ . $2 M = [42 - 2 2]$ Why this step? Each column's coordinates double, so each arrow's length doubles while its angle stays fixed.

Look at the blue original arrow $u$ and the longer yellow $2 u$ lying right on top of its direction — only the length changed.

Scale by $- 1$ . $(- 1) M = [- 2 - 1 1 - 1]$ Why this step? Multiplying by $- 1$ negates each coordinate, which points the arrow through the origin to the exact opposite side.

The red arrow $- u$ is the same length as blue $u$ but aims $18 0^{\circ}$ the other way — a reflection through the origin.

Verify: length of $u$ is $2^{2} + 1^{2} = 5$ ; length of $2 u = 4^{2} + 2^{2} = 20 = 25$ — exactly double ✓. Length of $- u = (- 2)^{2} + (- 1)^{2} = 5$ — unchanged ✓.

Cell H — real-world word problem (units matter)

Worked example Example 7 — doubling a price list (Cell H)

A shop stores weekly sales in a matrix $S$ where rows are two branches and columns are (units sold, price per unit in ₹): $S = [30502015]$ A festival doubles prices (units stay the same for now). But prices are only in column 2. Someone naively writes $2 S$ . What goes wrong, and what is the correct fix?

Forecast: does the whole matrix get scaled, or only one column? Think about what "double the prices" actually touches.

First see what $2 S$ would give. $2 S = [601004030]$ Why this step? Scalar multiplication is all-or-nothing — it cannot touch just one column. So $2 S$ wrongly doubles the units sold too.
Correct model: only column 2 is a price. Scaling a single column is not scalar multiplication of $S$ ; you scale that column separately. $S_{new} = [30504030]$ Why this step? The physical quantity being doubled (price, in ₹) lives only in column 2, so only those entries change.

Verify: units column unchanged: $30, 50$ — same as before ✓. Price column doubled: $20 \to 40$ , $15 \to 30$ ✓. Lesson: scalar multiplication scales everything; use it only when the whole grid is the same kind of quantity being scaled by the same factor.

Cell I — the exam twist: work the rule backwards

Worked example Example 8 — solve for the unknown (Cell I)

Find the scalar $k$ and then the matrix $X$ such that $k [3 - 6] = [12 - 24], 2 X = [86 - 4 0] .$

Forecast: in part 1 you know the before and after, so $k$ is a ratio. In part 2 you undo a doubling — what scalar undoes $2$ ?

Find $k$ : every entry was multiplied by the same $k$ , so compare any one entry. $k \cdot 3 = 12 ⟹ k = \frac{12}{3} = 4$ Why this step? Because $k$ is one number applied everywhere, one equation determines it.
Consistency check with the other entry. $4 \cdot (- 6) = - 24 ✓$ Why this step? If the two entries gave different $k$ , no single scalar would work — the check guards against that.
Solve for $X$ : multiply both sides by $\frac{1}{2}$ (the scalar that undoes $2$ ). $X = \frac{1}{2} [86 - 4 0] = [43 - 2 0]$ Why this step? $\frac{1}{2} \cdot 2 = 1$ , and $1 \cdot X = X$ (Cell E), so scaling by $\frac{1}{2}$ isolates $X$ .

Verify: plug back — $2 [43 - 2 0] = [86 - 4 0]$ ✓ and $4 [3 - 6] = [12 - 24]$ ✓.

Recall

Recall Which scenarios collapse or preserve the matrix?

$0 \cdot A$ gives the zero matrix; $1 \cdot A$ leaves $A$ unchanged. $0 \cdot A$ gives ::: the zero matrix $O$ (same shape, all zeros) $1 \cdot A$ gives ::: $A$ itself, unchanged

Recall Geometric effect of a negative scalar on a matrix's columns?

Reflect each column-arrow through the origin (same length, opposite direction). Effect of multiplying by $- 1$ ::: each column-arrow flips $18 0^{\circ}$ , length unchanged