Matrices & Systems — Practice
Strip the letters from a system of equations and what is left is a grid of numbers that can be solved mechanically.
What these questions 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
Give the dimensions of a matrix with 3 rows and 5 columns, in the form rows x columns.
Worked answer
3x5
R
o
w
s
f
i
r
s
t
,
a
l
w
a
y
s
.
Problem 2
Add [1234] and [5012]. Give the top-left entry.
Worked answer
6
1+5 = 6
Problem 3
Multiply [2314] by 5. Give the bottom-right entry.
Worked answer
20
4 × 5 = 20
Problem 4
Find the determinant of [3124].
Worked answer
10
3(4)-1(2) = 12-2 = 10
Problem 5
Find the determinant of [2613].
Worked answer
0
2(3)-6(1) = 6-6 = 0
Problem 6
Can a 2 × 3 matrix multiply a 2 × 4 matrix?
- No — the inside numbers disagree
- Yes, giving 2 × 4
- Yes, giving 3 × 2
- Only if both are square
Worked answer
No — the inside numbers disagree
3 ≠ 2, so the rows and columns cannot pair up
Problem 7
A 2 × 3 matrix multiplies a 3 × 4 matrix. Give the dimensions of the result, as rows x columns.
Worked answer
2x4
T
h
e
o
u
t
s
i
d
e
n
u
m
b
e
r
s
g
i
v
e
t
h
e
a
n
s
w
e
r
.
Problem 8
Find the top entry of [1230][45].
Worked answer
14
1(4)+2(5) = 14
Problem 9
Find the bottom entry of the same product.
Worked answer
12
3(4)+0(5) = 12
Problem 10
Is AB always equal to BA for matrices?
- No
- Yes
- Only for 2 × 2
- Only when both are invertible
Worked answer
No
Matrix multiplication depends on order
Often only one of the two products even exists
Problem 11
A 2 × 2 matrix has determinant 0. How many inverses does it have?
Worked answer
0
Zero determinant means no inverse exists
Problem 12
For [4726], find the determinant.
Worked answer
10
4(6)-7(2) = 24-14 = 10
Problem 13
For the system 2x+3y=7, 4x+6y=1, find the determinant of the coefficient matrix.
Worked answer
0
2(6)-3(4) = 0
So there is no unique solution
Problem 14
A system has coefficient determinant 0 and the equations are not multiples of each other. How many solutions?
- None
- Exactly one
- Infinitely many
- Two
Worked answer
None
The lines are parallel and distinct
Parallel lines never meet
Problem 15
What is the top-left entry of the 2 × 2 identity matrix?
Worked answer
1
[1001]
Problem 16
Find k so that [k436] has no inverse.
Worked answer
2
6k-12 = 0
k = 2