Matrices

A matrix is a series of rows and columns used to display numbers. ACT questions about matrices will generally be about performing fundamental math operations such as addition, subtraction, or multiplication between matrices.

Is It Possible?

A key note is that not all matrices can be added, subtracted, or multiplied.

• Two matrices can only be added or subtracted if they have the same dimensions
• Two matrices can only be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix

Adding and Subtracting

Adding and subtracting matrices is quite simple. Since the matrices have to have the same dimensions, each number has a “twin” that shares the same space in the opposite matrix.

Matrix addition means adding each pair of numbers in the same places in their respective matrices. Subtraction means subtracting pairs.

The resulting matrix will have the same dimensions as each of the original matrices.

Multiplying

Multiplying matrices is a bit more complicated. Matrix multiplication is used to find the dot product. This is sometimes the term used on the ACT to imply matrix multiplication.

As stated above, matrices can only be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix. The top left number in the dot product is equal to the sum of the numbers in the first row of the first matrix multiplied, in order, by the numbers in the first column of the second matrix.

The next number down in the dot product’s first column is equal to the numbers from the second row of the first matrix multiplied, in order, by the numbers in the first column of the second matrix.

This is then repeated until there are no more rows left in the first matrix. The process is then repeated again for the next column in the second matrix.

The final dot product will have the same number of rows as the first matrix and the same number of columns as the second matrix.