One Number Acting on an Entire Matrix

01 / Read before calculating

What is this asking us to do?

3M = 3 × [[1, 2], [3, 4]]

Read aloudThree multiplies the whole matrix M. Multiply each of its four entries by 3, keeping every answer at the same row and column address.

Objects in the expression

3

One scalar: a single number that will multiply each entry of M.

M

A whole matrix, here arranged in two rows and two columns. M is the object being scaled; it is not one of its entries.

1, 2, 3, 4

The four entries of M. Each has an address: row 1 contains 1 and 2; row 2 contains 3 and 4.

3M

Scalar multiplication: one number applied to a whole matrix. This differs from multiplying two matrices together.

02 / Visual model

One value, four destinations

03 / Full working

Walk through one example

Find 3M for M = [[1, 2], [3, 4]]. Keep the entries in their original positions as you calculate.

1. Name the objects3 × M, with M = [[1, 2], [3, 4]]

   The 3 is one scalar. M is one 2 × 2 matrix. Because the left object is a number, there is no row–column pairing to choose.

2. Keep each address3M = [[3 × 1, 3 × 2], [3 × 3, 3 × 4]]

   Copy the two-row, two-column layout. At every address, multiply the entry already there by 3. For example, the row 2, column 1 entry becomes 3 × 3.

3. Finish the arithmetic3M = [[3, 6], [9, 12]]

   Four input entries give four output entries. Scalar multiplication changes their values, not their locations or the shape of the matrix.

Result3 × [[1, 2], [3, 4]] = [[3, 6], [9, 12]].

04 / A tempting wrong turn

Where a reading can go astray

Multiply the number by just the first entry, or treat 3M as if two matrices need a row–column rule.

The brackets describe one matrix, and 3 is a scalar, not another matrix. Every entry receives the scalar once. If the expression were MN with both M and N matrices, it would be a different instruction with a shape check and matched row–column products. An arithmetic slip, such as writing 3 × 4 = 11, is different from misreading which operation the expression requests.

05 / Independent practice

Try these before opening the answers

The questions move from recognizing a symbol to applying and transferring the idea. Each answer shows the reasoning.

1. recognize

   P and Q are matrices. Which expression asks for scalar multiplication: 5P or PQ? What clue settles it?

   Show a hint

   Name the object to the left of P in each expression.

   Reveal worked answer

   5P asks for scalar multiplication; PQ asks for matrix multiplication.

   In 5P, 5 is one number, so it multiplies every entry of P.

   In PQ, both named objects are matrices. Its entries require the matrix-product rule, assuming their shapes are compatible.

2. recognize

   M has 2 rows and 3 columns, and s is a scalar. What is the shape of sM? If the entry of M at row 2, column 3 is m₂₃, what is the entry at that address in sM?

   Show a hint

   The scalar changes each value. Does it add or remove any addresses?

   Reveal worked answer

   sM is 2 × 3, and its row 2, column 3 entry is s × m₂₃.

   M has six addresses arranged in 2 rows and 3 columns; scaling leaves those addresses in place.

   At each address, multiply the existing entry by s. At (2, 3), that gives s × m₂₃.

3. apply

   Calculate −2 × [[3, −1], [0, 4]].

   Show a hint

   Write −2 times each of the four entries before simplifying.

   Reveal worked answer

   [[-6, 2], [0, -8]]

   Keep the two rows: [[−2 × 3, −2 × (−1)], [−2 × 0, −2 × 4]].

   The products are −6, 2, 0, and −8 in their original positions, giving [[−6, 2], [0, −8]].

4. apply

   Calculate (1/2) × [[6, 0], [−4, 10]].

   Show a hint

   Halve every entry, including zero and the negative entry.

   Reveal worked answer

   [[3, 0], [−2, 5]]

   At each address, multiply by 1/2: [[6/2, 0/2], [−4/2, 10/2]].

   Simplify without moving entries: [[3, 0], [−2, 5]].

5. transfer

   Let k(t) = 2t − 1 and P = [[1, 0], [2, −3]]. Find k(2)P. What would k(t)P mean before a value of t is chosen?

   Show a hint

   Evaluate k(2) first. For the symbolic part, leave k(t) as one factor at each address.

   Reveal worked answer

   k(2)P = [[3, 0], [6, −9]]. Before choosing t, k(t)P = [[k(t), 0], [2k(t), −3k(t)]].

   k(2) = 2 × 2 − 1 = 3, so the instruction becomes 3P.

   Multiply each entry of P by 3: [[3 × 1, 3 × 0], [3 × 2, 3 × (−3)]] = [[3, 0], [6, −9]].

   With t unspecified, use the same factor k(t) at every address: [[k(t), 0], [2k(t), −3k(t)]].

6. transfer

   In complex-number work, i² = −1. Compare iP with pᵢⱼ, where P = [pᵢⱼ]. What job does i do in each expression? If P = [[1, 2]], what is iP?

   Show a hint

   Look for the position of i: outside the brackets or down in an entry's address.

   Reveal worked answer

   In iP, i is the imaginary-unit scalar, so iP = [[i, 2i]]. In pᵢⱼ, i is the row index, not a multiplier.

   The statement i² = −1 declares i to be the imaginary unit in this context. Placed before P, it scales each entry; this gives [[i × 1, i × 2]] = [[i, 2i]]. The output has complex entries.

   In pᵢⱼ, subscripts identify an entry at row i and column j. That i names a position. The same printed letter can have a different role in a different context.

06 / Take it forward

Where this appears again

A single factor often scales an entire table of numbers. Later, Av = λv says that one scalar λ multiplies every component of a vector v; the left side, Av, is a matrix–vector product, while the right side, λv, is scalar multiplication. A letter's role comes from the expression and its stated context. In complex-number work, i may be the imaginary unit (i² = −1), so iM would scale each entry by i if complex entries are allowed. In mᵢⱼ, the subscript i is a row index instead: it selects an address and is not a number placed outside the matrix. Do not decide from the letter alone.