Matrices & Systems — Challenge
Strip the letters from a system of equations and what is left is a grid of numbers that can be solved mechanically.
What these problems cover
These questions use 9 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- System — Two or more equations that must hold at the same time.
- Parallel — Same slope, different intercept.
- Diagonal — A segment joining two non-adjacent vertices.
- Matrix — A rectangular array of numbers, described by its rows then columns.
- Dimensions — Rows × columns. A 2 × 3 matrix has two rows and three columns.
- Identity matrix — The square matrix with ones down the diagonal; it leaves everything unchanged.
- Determinant — A single number computed from a square matrix; zero means it has no inverse.
Problem 1
If A is 3 × 2 and B is 2 × 3, what are the dimensions of AB?
- 3 × 3
- 2 × 2
- 3 × 2
- 2 × 3
- Undefined
Worked answer
3 × 3
Write the shapes side by side: (3 × 2)(2 × 3). The inside numbers are both 2, so the product exists. The outside numbers give the answer: 3 × 3. Note that BA would be 2 × 2 — a different size entirely.
Problem 2
Find the determinant of [2143].
- 2
- 10
- -2
- 6
- 0
Worked answer
2
The determinant of a 2×2 matrix is ad - bc. = 2(3) - 1(4) = 6 - 4. = 2. Being non-zero, the matrix has an inverse.
Problem 3
For which values of k does [k28k] have no inverse?
- k = ± 4
- k = ± 2
- k = 4 only
- k = 16
- k = 0
Worked answer
k = ± 4
No inverse means the determinant is zero. det = k2 - 16 = 0. So k2 = 16 and k = ± 4. Both work, because the determinant depends on k only through its square.
Problem 4
A 2 × 2 matrix A has det A = 3. What is det(2A)?
- 12
- 6
- 3
- 24
- 4
Worked answer
12
Multiplying by 2 doubles every entry, so it doubles both rows. Each doubled row multiplies the determinant by 2, and there are two rows. So det(2A) = 22 · 3 = 12. Geometrically, doubling both directions quadruples an area.
Problem 5
Find the top-left entry of the inverse of [1234].
- -2
- 2
- 4
- -12
- 1
Worked answer
-2
The determinant is 1(4) - 2(3) = -2. The inverse is 1det[d-b-ca] = -12[4-2-31]. The top-left entry is -12 × 4 = -2.
Problem 6
For square matrices A and B, the statement AB = BA is
- sometimes true
- always true
- never true
- true only for 1 × 1 matrices
- true only when det A = 0
Worked answer
sometimes true
It fails in general — matrix multiplication depends on the order. But it does hold in particular cases: any matrix commutes with the identity, and with its own inverse. So "sometimes" is the only accurate answer.
Problem 7
A system of two linear equations has coefficient determinant 0, and the second equation is exactly twice the first. How many solutions?
- Infinitely many
- None
- Exactly one
- Exactly two
- Cannot be determined
Worked answer
Infinitely many
A zero determinant rules out a unique solution. That leaves two cases: parallel and distinct lines, or the same line twice. Here the second equation is a multiple of the first, so they describe the same line and every point on it is a solution.
Problem 8
What is the determinant of the 2 × 2 identity matrix?
- 1
- 0
- 2
- 4
- -1
Worked answer
1
The identity is [1001]. det = 1(1) - 0(0). = 1, which fits its meaning: the identity leaves everything unchanged, so it stretches area by a factor of one.