If A = $\begin{pmatrix} 2 & 1 \\ 2 & 3 \\ 1 & 2 \end{pmatrix}$ and B = $\begin{pmatrix} 3 & 2 \\ 4 & 2 \end{pmatrix}$. Find AB

Analysis

To find the product of two matrices $AB$, we must multiply the rows of matrix $A$ by the columns of matrix $B$.

• Matrix $A$ is a $3 \times 2$ matrix.
• Matrix $B$ is a $2 \times 2$ matrix.
• The resulting matrix $AB$ will be a $3 \times 2$ matrix.

Concept

For matrix multiplication, we multiply the elements of the row of the first matrix by the corresponding elements of the column of the second matrix and sum them up. If $C = AB$, then $C_{ij} = (\text{Row } i \text{ of } A) \cdot (\text{Column } j \text{ of } B)$.

Solution

Given: $A = \begin{pmatrix} 2 & 1 \\ 2 & 3 \\ 1 & 2 \end{pmatrix}$, $B = \begin{pmatrix} 3 & 2 \\ 4 & 2 \end{pmatrix}$

Step 1: Calculate the first row of the result

• Row 1, Col 1: $(2 \times 3) + (1 \times 4) = 6 + 4 = 10$
• Row 1, Col 2: $(2 \times 2) + (1 \times 2) = 4 + 2 = 6$

Step 2: Calculate the second row of the result

• Row 2, Col 1: $(2 \times 3) + (3 \times 4) = 6 + 12 = 18$
• Row 2, Col 2: $(2 \times 2) + (3 \times 2) = 4 + 6 = 10$

Step 3: Calculate the third row of the result

• Row 3, Col 1: $(1 \times 3) + (2 \times 4) = 3 + 8 = 11$
• Row 3, Col 2: $(1 \times 2) + (2 \times 2) = 2 + 4 = 6$

Putting it all together: $AB = \begin{pmatrix} 10 & 6 \\ 18 & 10 \\ 11 & 6 \end{pmatrix}$

Conclusion

Comparing our result with the given options, we see that it matches option D.

Exam Tip

When multiplying matrices, always remember the phrase "Row by Column". Use your left index finger to trace the row of the first matrix and your right index finger to trace the column of the second matrix simultaneously. This prevents you from skipping elements or getting confused during the calculation.