Matrix multiplication
Matrix multiplication combines two matrices by taking dot products of rows and columns. It is defined only when the number of columns in the first matrix equals the number of rows in the second.
Matrix multiplication is the operation that combines two matrices into a new matrix. Unlike addition, which simply pairs matching entries, multiplication pairs rows of the first matrix with columns of the second: each entry in the product is the dot product of a row from the first matrix and a column from the second — multiply the corresponding entries and add them up.
The dimension rule decides everything. Multiplying an m × n matrix by an n × p matrix is defined only because the inner dimensions match (n = n), and the product has the outer dimensions, m × p. A 2 × 3 matrix times a 3 × 2 matrix gives a 2 × 2 result; a 2 × 3 times a 2 × 3 is undefined. Standardized tests love this rule because it can be checked without computing anything.
To compute an entry, trace the row and column that meet there. If row 1 of the first matrix is (1, 2, 3) and column 1 of the second is (4, 5, 6), the entry in row 1, column 1 of the product is 1 × 4 + 2 × 5 + 3 × 6 = 32. Note that matrix multiplication is not commutative — AB and BA are generally different, and one may not even be defined when the other is.
On the ACT math section, matrix multiplication questions usually ask one of two things: whether a product is defined and what size it will be, or the value of a single entry in a small product. Checking inner dimensions first and computing only the entry you need are the fastest paths to the answer.
Key takeaways
- Each entry of a matrix product is the dot product of a row from the first matrix and a column from the second.
- An m × n matrix times an n × p matrix is defined only when the inner dimensions match, and the result is m × p.
- Matrix multiplication is not commutative: AB generally does not equal BA.
- ACT questions typically test the dimension rule or ask for a single entry of a small product.