Messing with Matrix Questions on the ACT Mathematics Test

By Lisa Zimmer Hatch, Scott A. Hatch

Every once in a while the ACT may slip a matrix problem into the Mathematics Test. If you see one, don’t panic. They’re easy to deal with when you review the approach.

A matrix is simply an array of values. Although you can perform several operations with matrices, the ACT will likely ask you to multiply them. Sometimes the problem will be as elementary as multiplying a matrix by one value to form another matrix. (This operation is called scalar multiplication, but you don’t really need to know that.)

The process is a little more complex when a question asks you to multiply two matrices. This activity requires you to multiply the values in the first row of the first matrix by the values in the first row of the second. The simplest version is finding the product of a one-row matrix and a one-column matrix. You add the first number in the first matrix by the first number in the second matrix, and then add the product of the second value in the first and second matrices, to which you then add the product of the third values in each of the matrices. It looks like this:

image0.png

When the matrices have more than one row or column, you apply the same approach and get more than one value. The resulting matrix product will always have the same number of rows as the first matrix and the same number of columns as the second, like this:

image1.png

Here are the steps to arriving at a solution:

  1. Add the products of the first row of the first matrix and the first column of the second matrix as we demonstrated above to get the first value in the matrix product (35).

  2. Add the products of the first row in the first matrix and the second column of the second matrix to find the second value in the matrix product (55):

    image2.png

  3. Add the products of the second row of the first matrix and the first column of the second matrix to find the third value in the matrix product (10):

    image3.png

  4. Add the products of the second row of the first matrix to the second column of the second matrix to find the fourth value in the matrix product (20):

    image4.png

You can only multiply matrices if the number of columns in the first equals the number of rows in the second.