6.1 Matrices & Determinants
Key Takeaways
- An m×n matrix has m rows and n columns; addition needs equal order, while AB needs columns of A equal to rows of B.
- det([a b; c d]) = ad − bc; a 2×2 matrix is invertible iff its determinant is nonzero.
- For 3×3 matrices, expand along a row/column (cofactor expansion) or use the rule of Sarrus for a quick numeric check.
- A⁻¹ = (1/det A) · adj A; for 2×2, swap diagonals and negate off-diagonals, then divide by det A.
- Cramer’s rule solves Ax = b via xᵢ = det(Aᵢ)/det(A) when det(A) ≠ 0—fast for 2×2 systems on timed MCQs.
Why Matrices Appear on the PAF CAE Maths Paper
Matrices and determinants sit in the FSc Pre-Engineering core and show up regularly in Pakistan Air Force College of Aeronautical Engineering (CAE) and related engineering-branch academic maths banks. Coaching and candidate reports often describe a computer-based maths paper of roughly 50 MCQs in about 25 minutes (confirm your cycle’s slip on joinpaf.gov.pk—exact counts are not always published the same way every year). That pacing leaves about 30 seconds per question, so matrix items reward instant recall of order rules, determinant shortcuts, and whether an inverse exists—not long Gaussian eliminations by hand.
Think of a matrix as a rectangular array of numbers used to store linear data compactly: coefficients of simultaneous equations, transformation rules, or tabulated engineering quantities. On the exam you will be asked to add, multiply, find determinants, decide invertibility, and apply Cramer’s rule to small systems.
Order, Types, and Notation
An m × n matrix has m rows and n columns. Element aᵢⱼ sits in row i, column j (1-based indexing in FSc texts).
| Type | Condition | Exam cue |
|---|---|---|
| Row matrix | 1 × n | Single row of data |
| Column matrix | m × 1 | Single column (often a solution vector) |
| Square matrix | m = n | Only square matrices have determinants/inverses in the usual school sense |
| Diagonal | Square; aᵢⱼ = 0 for i ≠ j | Off-diagonals zero |
| Identity Iₙ | 1s on diagonal, 0 elsewhere | AI = IA = A |
| Zero / null O | All entries 0 | A + O = A |
| Symmetric | Aᵀ = A | Mirror across main diagonal |
| Skew-symmetric | Aᵀ = −A | Diagonal must be zero |
Transpose: (Aᵀ)ᵢⱼ = Aⱼᵢ. Useful identities: (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, (AB)ᵀ = BᵀAᵀ (order reverses).
Matrix Operations
Addition / subtraction. Same order required. Add or subtract corresponding entries. Commutative: A + B = B + A.
Scalar multiplication. kA multiplies every entry by k.
Multiplication. If A is m × n and B is n × p, then AB is m × p with
(AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ.
Multiplication is associative and distributive, but generally not commutative: AB may exist while BA does not, or both exist yet AB ≠ BA.
Worked — multiplication.
Let A = [[1, 2], [0, −1]] (2×2) and B = [[3, 0], [1, 4]] (2×2).
AB = [[1·3+2·1, 1·0+2·4], [0·3+(−1)·1, 0·0+(−1)·4]] = [[5, 8], [−1, −4]].
BA = [[3·1+0·0, 3·2+0·(−1)], [1·1+4·0, 1·2+4·(−1)]] = [[3, 6], [1, −2]].
So AB ≠ BA—classic MCQ trap if you assume commutativity.
Worked — order check. A is 2×3, B is 2×3 → A + B is fine, but AB is undefined (3 ≠ 2). BA would need B’s columns (3) equal to A’s rows (2)—also undefined. Always check inner dimensions first.
Determinants: 2×2 and 3×3
Only square matrices have a determinant (a scalar).
2×2 formula. For A = [[a, b], [c, d]],
det(A) = |A| = ad − bc.
Worked. |[[4, −2], [3, 5]]| = 4·5 − (−2)·3 = 20 + 6 = 26.
3×3 cofactor expansion along row 1:
For A = [[a₁₁, a₁₂, a₁₃], [a₂₁, a₂₂, a₂₃], [a₃₁, a₃₂, a₃₃]],
|A| = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁).
(Signs along row 1 are +, −, +.)
Worked — 3×3.
A = [[2, 1, 0], [−1, 3, 4], [0, 2, 1]].
|A| = 2(3·1 − 4·2) − 1((−1)·1 − 4·0) + 0(...) = 2(3 − 8) − 1(−1 − 0) + 0 = 2(−5) − (−1) = −10 + 1 = −9.
Properties that save time on MCQs:
| Property | Consequence |
|---|---|
| Swap two rows/columns | Determinant changes sign |
| Two identical rows/columns | det = 0 |
| Factor k from one row/column | det multiplies by k |
| Row replaced by row + k·(another row) | det unchanged |
| AB | |
| Aᵀ | |
| kA |
If det(A) = 0, A is singular (not invertible). If det(A) ≠ 0, A is nonsingular (invertible).
Inverse of a Matrix (Introduction)
For square A, the inverse A⁻¹ satisfies A A⁻¹ = A⁻¹ A = I.
General formula: A⁻¹ = (1/|A|) · adj(A), where adj(A) is the adjoint (transpose of the cofactor matrix), provided |A| ≠ 0.
2×2 quick rule. If A = [[a, b], [c, d]] and Δ = ad − bc ≠ 0,
A⁻¹ = (1/Δ) [[d, −b], [−c, a]].
Swap the main-diagonal entries, negate the off-diagonal entries, divide by the determinant.
Worked — inverse.
A = [[3, 1], [2, 4]]. Δ = 12 − 2 = 10.
A⁻¹ = (1/10) [[4, −1], [−2, 3]] = [[0.4, −0.1], [−0.2, 0.3]].
Check: A A⁻¹ = I₂ (spot-check first row: 3·0.4 + 1·(−0.2) = 1.2 − 0.2 = 1; 3·(−0.1) + 1·0.3 = 0).
Exam trap: If Δ = 0, stop—there is no inverse. Do not “force” the 2×2 swap formula.
Cramer’s Rule (Idea and 2×2 Practice)
For a linear system A x = b with square coefficient matrix A and |A| ≠ 0, Cramer’s rule says each unknown is a ratio of determinants:
xᵢ = |Aᵢ| / |A|,
where Aᵢ is A with column i replaced by the constant vector b.
Worked — 2×2 system.
Solve:
2x + y = 5
3x − y = 4
A = [[2, 1], [3, −1]], b = [[5], [4]].
|A| = −2 − 3 = −5.
A₁ (replace col 1): [[5, 1], [4, −1]], |A₁| = −5 − 4 = −9 → x = (−9)/(−5) = 9/5.
A₂ (replace col 2): [[2, 5], [3, 4]], |A₂| = 8 − 15 = −7 → y = (−7)/(−5) = 7/5.
Check: 2(9/5) + 7/5 = 18/5 + 7/5 = 25/5 = 5 ✓.
For 3×3 systems the same idea applies, but arithmetic grows—on a ~30 s/question paper, expect mostly 2×2 Cramer or a single 3×3 determinant, not a full three-unknown grind unless numbers are tiny.
Exam Strategy Under Time Pressure
- Read the order before multiplying—undefined products are free marks if you catch them.
- Compute 2×2 dets in one line (ad − bc); expand 3×3 along a row with zeros when possible.
- Invertibility test is just “is det zero?”—do not build the full inverse unless asked.
- Cramer for 2×2: three tiny determinants beat substitution when options are fractions.
- Watch AB vs BA and |kA| = kⁿ|A| traps.
Master these mechanics and you clear a high-frequency FSc cluster that feeds straight into later linear-algebra ideas used in engineering coursework—exactly the style of recognition the CAE maths paper rewards.
If A is a 2×3 matrix and B is a 3×2 matrix, which statement is correct?
What is the determinant of [[5, −3], [2, 4]]?
A square matrix A has det(A) = 0. Which conclusion follows?
Using Cramer’s rule on 3x + y = 7 and x − 2y = 0, the value of y is: