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MAT423 · chapter 1 · Question 1

Matrices

6 recipes. Each one: read the recipe, watch it, drag it, then do it yourself.

RECIPERead a matrix

  1. Size = rows × columns (rows go across ↔, columns go down ↕). 1 2;3 0;-1 4 has 3 rows and 2 columns → 3 × 2 ("3 by 2").
  2. Entry aij = the number in row i, column j. In the matrix above, a21 = 3 (row 2, column 1).
  3. Equal matrices: same size AND every matching entry is equal. So 2 1;3 x = y z;3 5 gives x = 5, y = 2, z = 1.
  4. Main diagonal = top-left to bottom-right (a11, a22, a33 …). Most matrix "types" are about this line.
Senang ceritaAlways say ROWS first, then COLUMNS. "RC", like "Rumah dulu, Cawan kemudian".

TAP ITThe matrix types

DO ITName that matrix

REMEMBERMemory hooks

🔺
Upper triangular = zeros live DOWNstairs (below the diagonal). Lower = zeros live UPstairs.
🪞
Symmetric = the diagonal is a mirror: a12 = a21, a13 = a31 … so A = Aᵀ.
TRAPTriangular, diagonal, identity and symmetric must all be square (n × n).

PROVE IT3 quick checks

RECIPEThree operations

  1. Add / subtract: only if the sizes are the SAME. Add matching boxes. 4 3;5 2 + 2 -1;8 0 = 6 2;13 2
  2. Scalar × matrix (kA): multiply EVERY entry by k. 22 4 3;-7 5 0 = 4 8 6;-14 10 0
  3. Size check for AB: write (m × p)(p × n). The INSIDE numbers must match. The answer size is the OUTSIDE numbers m × n.
  4. Each box of AB = (row i of A) · (column j of B): multiply pairs, then add. Row 1 · column 1 → box (1,1).
Senang ceritaMultiplying = your left finger walks ACROSS a row of A, your right finger walks DOWN a column of B. Multiply the pairs you touch, add them up.

WATCH ITNotes Example 14, step by step

TAP ITRow × column, live

TRY ITCan I multiply these sizes?

CALCULATORCheck AB on your calculator

CASIO fx-570EX / 991EX (ClassWiz)MENU 4 Matrix → MatA → rows 2= columns 2= → type each number + = → AC. Do the same for MatB (OPTN 1 Define Matrix → MatB).
Then: OPTN MatA × OPTN MatB =

CASIO fx-570ES PLUS / 991ES PLUSMODE 6 → 1 MatA → pick size → type numbers → AC. Matrix menu = SHIFT4: 3 MatA, 4 MatB. So: SHIFT43 × SHIFT44 =
Use it to CHECK. Marks come from the working you write, so still show row × column.

DO ITPractice

REMEMBERMemory hooks

🚫
AB ≠ BA. Order matters! Changing the order changes the answer (sometimes BA does not even exist).
➗
No dividing matrices. To "divide" you multiply by an inverse (Recipe 4 and 5).
TRAPDo not multiply box-by-box like adding. 1 2;3 41 0;0 2 is NOT 1 0;0 8.

PROVE IT3 quick checks

RECIPEFlip it

  1. Row 1 of A becomes column 1 of Aᵀ, row 2 becomes column 2, … 5 4;2 1;6 7ᵀ = 5 2 6;4 1 7 (3×2 becomes 2×3).
  2. Given Kᵀ and you need K? Just transpose again: (Kᵀ)ᵀ = K. (Past papers do this a lot!)
  3. The 5 rules: (Aᵀ)ᵀ = A · (A + B)ᵀ = Aᵀ + Bᵀ · (kA)ᵀ = kAᵀ · (AB)ᵀ = BᵀAᵀ · AAᵀ and AᵀA are symmetric.
Socks and shoesYou put on socks THEN shoes. To undo, shoes come off FIRST. (AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹: the order reverses.

WATCH ITNotes Example 16

CALCULATORTranspose button

ClassWizOPTN ▼ Transposition → OPTN MatA ) =
ES PLUSSHIFT48 (Trn) SHIFT43 ) =

DO ITPractice

REMEMBERMemory hooks

TRAP(AB)ᵀ ≠ AᵀBᵀ. The notes show it with numbers: BᵀAᵀ = (2 −2; −3 2) but AᵀBᵀ = (3 −1; −5 1).

PROVE IT3 quick checks

RECIPETukar, tolak, bahagi

A = a b;c d  ⟹  A⁻¹ = 1/ad − bc d -b;-c a
  1. Find |A| = ad − bc (down-right diagonal minus up-right diagonal). If it is 0 → no inverse ("singular"). Stop and say so.
  2. Tukar: swap a and d (the main diagonal).
  3. Tolak: put a minus sign on b and c (they stay in place).
  4. Bahagi: divide every entry by |A|. Leave fractions as fractions.
  5. Check (optional): A·A⁻¹ = I.
Senang cerita"Tukar the diagonal, tolak the other two, bahagi with the det."

WATCH ITNotes Example 17

DRAG ITSee what an inverse does

A matrix moves every point of the plane. A⁻¹ moves them back. When det = 0 the plane gets squashed onto a line, and you can't un-squash it, so there's no inverse.

CALCULATORInverse button (to check)

ClassWizPut the matrix in MatA → OPTN MatA x⁻¹ = (it shows fractions). If |A| = 0 you get "Math ERROR".
ES PLUSSHIFT43 x⁻¹ =

DO ITPractice

REMEMBERThe inverse rules

(A⁻¹)⁻¹ = A · (Aᵏ)⁻¹ = (A⁻¹)ᵏ · (cA)⁻¹ = (1/c)A⁻¹ · (Aᵀ)⁻¹ = (A⁻¹)ᵀ · (AB)⁻¹ = B⁻¹A⁻¹ (socks and shoes again)
TRAPOnly b and c change sign. a and d swap but keep their own signs.

PROVE IT3 quick checks

RECIPEPeel the onion

  1. Find the unknown (C, L or Zᵀ) and look at what is on its LEFT and on its RIGHT.
  2. A matrix on the left end leaves by multiplying its inverse on the LEFT of both sides. A matrix on the right end leaves by multiplying its inverse on the RIGHT of both sides.
  3. Remove the OUTSIDE ones first. Write the formula, e.g. CAB = M ⟹ C = MB⁻¹A⁻¹.
  4. The inverse of X⁻¹ is X, so "K⁻¹ on the right" is removed by multiplying K on the right.
  5. Work out the inverses you need (Recipe 4), then multiply two matrices at a time, left to right.
  6. If the unknown was Zᵀ, transpose your answer at the end to get Z.
Senang ceritaMatrices cannot jump over each other. Whoever is on the left goes out through the left door; whoever is on the right goes out through the right door.

PLAY ITPeel the real past-year equations

WATCH ITJul 2025 Q1(a)(ii), full working

CALCULATORCheck the final answer

Store A in MatA, B in MatB, M in MatC. For C = MB⁻¹A⁻¹ type: OPTN MatC × OPTN MatB x⁻¹ × OPTN MatA x⁻¹ = (ES PLUS: use SHIFT4). Exactly the order of your formula.

DO ITPractice (endless)

REMEMBERMemory hooks

🚪
Left door, right door. Never move a matrix to the other side of the unknown.
TRAPWriting C = A⁻¹MB⁻¹ for CAB = M is the #1 mistake. A is on the RIGHT of C, so A⁻¹ goes on the right too.

PROVE IT3 quick checks

RECIPEUntangle, then multiply letters

  1. Remove the outside ᵀ or ⁻¹ first. (MN)ᵀ = P ⟹ MN = Pᵀ. (DE)⁻¹ = E⁻¹D⁻¹.
  2. Peel exactly like Recipe 5: (MN)N⁻¹ = M. D⁻¹(DE) = E. E(DE)⁻¹ = D⁻¹.
  3. Multiply the letter matrices row × column: p r;q st u;v w = pt+rv pu+rw;qt+sv qu+sw
  4. If they ask for Eᵀ or Q², do that last step (transpose / multiply by itself).
The 5 versions seen so far Feb 2026: (MN)ᵀ, N⁻¹ → M = PᵀN⁻¹ · Jul 2025: (DE)⁻¹, Eᵀ → D⁻¹ = E(DE)⁻¹ · Jul 2024: (FG)ᵀ, F⁻¹ → G = F⁻¹(FG) · Jan 2023: DE, D⁻¹ → E = D⁻¹(DE) then Eᵀ · Feb 2025: P = I → Q = P⁻¹Q, then Q².

WATCH ITFeb 2026 Q1(b)

DO ITSame method with numbers (so it can be checked)

REMEMBERMemory hooks

🧦
(DE)⁻¹ = E⁻¹D⁻¹, (MN)ᵀ = NᵀMᵀ: the order flips.
TRAPWhen you transpose (p q; r s) you get (p r; q s). Only q and r swap places.

PROVE IT3 quick checks

How to scoreQ1 is about 11 marks: (a)(i) 2m · (a)(ii) 5m · (b) 4m. Marks come from the steps: formula, each inverse, each product. Always write the rearranged formula first (e.g. C = MB⁻¹A⁻¹). That line alone earns a mark.