Some ACT problems require unfamiliar notation as well as algebra. Translate matrices, vectors and complex numbers into operations you already know. For counting and sequences, first decide whether order matters and whether the pattern is additive or multiplicative.
A reliable method
Use the definition
Matrix multiplication takes a row–column dot product, not elementwise products. Add vectors component by component. For complex numbers use i² = −1 when expanding products.
Distinguish selection from arrangement
Choosing two different people for two distinct roles has order. Choosing an unordered pair does not. In a probability calculation, count favourable and total outcomes using the same rule.
Identify a sequence’s pattern
An arithmetic sequence adds the same difference: aₙ = a₁ + (n − 1)d. A geometric sequence multiplies by the same ratio: aₙ = a₁r^(n − 1). The first term uses exponent zero.
Check dimensions and sign
A product of a 2×3 matrix and a 3×1 matrix is 2×1. A vector’s magnitude is nonnegative. In a complex product the i² term contributes a negative real amount.
Worked examples
What is the first entry of the product of matrix [[2,−1],[3,4]] and vector [5,2]?
- Use the first row [2,−1] and the column [5,2].
- Compute 2×5 + (−1)×2 = 8.
- For comparison, the second entry is 3×5 + 4×2 = 23. The output is [8,23].
Answer: 8
Six students are eligible for president and secretary. One person cannot hold both roles. How many assignments are possible?
- There are six choices for president.
- For each choice, five students remain eligible for secretary.
- Multiply 6 × 5 = 30. Do not divide by two: swapping the roles creates a different assignment.
Answer: 30 assignments
Now use the method yourself.
Mark this guide as read, then use the method in topic practice before the checkpoint.