How to Solve ACT® Math Questions
Tricks and Strategies

• Explanation

In a defined matrix product, the resulting matrix always has the same number of rows as the first matrix and the same number of columns as the second matrix.

In the given matrix product, the first matrix is a 2 × 2 matrix (2 rows, 2 columns) and the second is a 2 × 1 matrix (2 rows, 1 column), so the resulting matrix must be a 2 × 1 matrix (2 rows, 1 column).

Set up the matrix product $\left[\left[\begin{array}{cc}–2& \phantom{0}1\\ \phantom{0}2& –6\end{array}\right]\right]\cdot \left[\left[\begin{array}{c}–4\\ \phantom{0}3\end{array}\right]\right]$ and the resulting 2 × 1 matrix.

To calculate each entry in a particular row and column of the resulting matrix, pair entries from that row of the first matrix (moving left to right: $\to$with entries from that column of the second matrix (moving top to bottom: $\downarrow$).

Find the product of each pair, and then add all the products to find the result.

Eliminate choice E because the entry in the first row is not 11.

The given matrix product is equal to the matrix $\left[\left[\begin{array}{c}\phantom{–}\mathbf{11}\\ \mathbf{–}\mathbf{26}\end{array}\right]\right]$ To verify the result, see complete calculation.

Note: In matrix multiplication, the order of the matrices matters. For example, the given product $\left[\left[\begin{array}{cc}–2& \phantom{0}1\\ \phantom{0}2& –6\end{array}\right]\right]\cdot \left[\left[\begin{array}{c}–4\\ \phantom{0}3\end{array}\right]\right]$ is equal to $\left[\left[\begin{array}{c}\phantom{–}11\\ –26\end{array}\right]\right]$ , but the reordered product$\left[\left[\begin{array}{c}–4\\ \phantom{0}3\end{array}\right]\right]\cdot \left[\left[\begin{array}{cc}–2& \phantom{0}1\\ \phantom{0}2& –6\end{array}\right]\right]$is undefined..

(Choices A, B, and C) $\left[\left[\begin{array}{cc}\phantom{0}8& \phantom{0}\phantom{0}3\\ –8& –18\end{array}\right]\right]$ and $\left[\left[\begin{array}{cc}8& –4\\ 6& –18\end{array}\right]\right]$ cannot be equal to the given matrix product because they are 2 × 2 matrices. The matrices in the given product have dimensions of 2 × 2 and 2 × 1, so the result must be a 2 × 1 matrix.

(Choice E) $\left[\left[\begin{array}{c}\phantom{0}5\\ 10\end{array}\right]\right]$ may result from mistakenly subtracting (instead of adding) when calculating the entries of the resulting matrix.

Things to remember:

• In a matrix product, the resulting matrix must have the same number of rows as the first matrix and the same number of columns as the second matrix. For example:
  
  
• To calculate a matrix product, align the first and second matrices at the left and top edges of the resulting matrix, respectively. To calculate each entry, use the pattern below to multiply pairs of entries and add them.