IMO Practice Test — Matrices
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
medium
Find x and y if [x+y, x-y; 2, 3] = [6, 2; 2, 3]
x=4,y=2
x=2,y=4
x=4,y=-2
x=3,y=3
Explanation: Equating entries: x+y=6 and x-y=2. Adding gives 2x=8, so x=4 and y=2.
Question 2 of 6
medium
If 2[3 4; 5 6] + 3[1 2; 3 4] = [a b; c d], find a+b+c+d
44
48
52
66
Explanation: 2A = [6 8; 10 12] and 3B = [3 6; 9 12], so the sum is [9 14; 19 24]. Then a+b+c+d = 9+14+19+24 = 66.
Question 3 of 6
medium
If A = [1 2; 3 4] and B = [2 0; 1 2], find the matrix X such that 2X + A = B
[0.5 -1; -1 -1]
[0.5 1; -1 1]
[-0.5 -1; 1 -1]
[0.5 1; -1 -1]
Explanation: From 2X + A = B, 2X = B - A = [1 -2; -2 -2], so X = [0.5 -1; -1 -1].
Question 4 of 6
medium
For what value of k does k[2 4; 6 8] = [6 12; 18 24]?
2
3
4
5
Explanation: k×2=6→k=3; also 3×4=12, 3×6=18, 3×8=24 ✓
Question 5 of 6
medium
If A = [1 2; 3 4], B = [5 6; 7 8], then (A+B) − (A−B) = ?
B
2B
2A
Zero matrix
Explanation: (A+B)-(A-B)=A+B-A+B=2B
Question 6 of 6
medium
Find the value of a and b if [a+b, a-b; 4, 6] = [8, 2; 4, 6]
a=5,b=3
a=3,b=5
a=5,b=-3
a=6,b=2
Explanation: Equating entries: a+b=8 and a-b=2. Adding gives 2a=10, so a=5 and b=3.