How To Find The Product Of Matrices

Introduction When you first encounter matrix multiplication, it can feel like stepping into a new language—one that uses rows and columns instead of words and sentences. Yet, just as sentences are built from words, the product of two matrices is built from simple, repeatable operations. In this article we will demystify the process of finding the product of matrices, explain why the method works, and show you how to apply it confidently in algebra, computer graphics, economics, and beyond. By the end, you’ll not only know the mechanics but also understand the underlying logic that makes matrix multiplication a cornerstone of linear algebra.

Detailed Explanation

Matrix multiplication is not the same as multiplying numbers. Even so, instead, it combines vectors and arrays of numbers to produce a new matrix that captures linear transformations. The key rule governing the product is that the number of columns in the first matrix must equal the number of rows in the second matrix. If the first matrix is of size m × n (m rows, n columns) and the second matrix is of size n × p, the resulting product will be an m × p matrix. This requirement ensures that each entry in the resulting matrix can be formed by taking the dot product of a row from the first matrix with a column from the second matrix.

Why does this rule exist? Plus, think of a matrix as a compact representation of a system of linear equations. In real terms, multiplying two matrices effectively chains together two sets of linear transformations: the first matrix transforms a vector in n‑dimensional space, and the second matrix transforms the result into an m‑dimensional space. The product matrix therefore encodes the overall transformation from the original space directly to the final space. Without the matching dimension rule, the transformations could not be composed meaningfully.

Some disagree here. Fair enough.

Understanding the dimension rule is the first step toward mastering matrix multiplication. Once you internalize that the inner dimensions must align, the rest of the process becomes a systematic series of calculations that can be performed by hand or with software.

Step‑by‑Step or Concept Breakdown

To compute the product of two matrices, follow these logical steps:

1. Verify Compatibility

   • Check that the number of columns in the first matrix equals the number of rows in the second matrix.
   • Example: If A is 2 × 3 and B is 3 × 4, the product AB is defined and will be 2 × 4.

2. Set Up the Result Grid

   • The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
   • If A is m × n and B is n × p, then AB will be m × p.

3. Compute Each Entry

   • For each position (i, j) in the result, take the i‑th row of A and the j‑th column of B.
   • Multiply the corresponding elements pairwise and sum the products. This is the dot product of the row and column.
   • Mathematically:
     [ (AB){ij} = \sum{k=1}^{n} A_{ik} \times B_{kj} ]

4. Fill the Matrix

   • Repeat step 3 for every row i (from 1 to m) and every column j (from 1 to p).
   • Populate the result grid with each computed value.

5. Double‑Check Dimensions

   • Ensure the final matrix’s size matches the expected m × p dimensions.
   • Verify a few entries manually to catch arithmetic errors.

Why bullet points help: They break down a potentially intimidating process into bite‑size actions, making it easier to remember and apply, especially when working without a calculator.

Quick Reference Checklist

• Compatibility? → Columns of A = Rows of B
• Result size? → m × p
• Entry formula? → Row · Column sum
• Repeat for all rows/columns

Real Examples

Let’s apply the steps to concrete matrices.

Example 1: Multiply

[A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix} \quad (2 \times 3) ]

by

[ B = \begin{bmatrix} 7 & 8 \ 9 & 10 \ 11 & 12 \end{bmatrix} \quad (3 \times 2) ]

Step 1: Verify compatibility – A has 3 columns, B has 3 rows → OK.
Step 2: Result will be 2 × 2. Step 3: Compute each entry:

• (1,1): Row 1 of A = (1, 2, 3), Column 1 of B = (7, 9, 11) → 1·7 + 2·9 + 3·11 = 7 + 18 + 33 = 58
• (1,2): Row 1 of A with Column 2 of B = (8, 10, 12) → 1·8 + 2·10 + 3·12 = 8 + 20 + 36 = 64
• (2,1): Row 2 of A = (4, 5, 6) with Column 1 of B → 4·7 + 5·9 + 6·11 = 28 + 45 + 66 = 139
• (2,2): Row 2 of A with Column 2 of B → 4·8 + 5·10 + 6·12 = 32 + 50 + 72 = 154

Result:

[AB = \begin{bmatrix} 58 & 64 \ 139 & 154 \end{bmatrix} ]

Example 2: A case where multiplication is not possible.

If C is 3 × 2 and D is 4 × 5, the inner dimensions (2 vs. 4) do not

Example2 (continued):
If C is 3 × 2 and D is 4 × 5, the inner dimensions (2 vs. 4) do not match. Matrix multiplication is undefined in this case because the number of columns in C (2) does not equal the number of rows in D (4). This incompatibility means there is no way to align the rows and columns for the dot product operation required to compute entries in the product matrix Simple, but easy to overlook. Took long enough..

Example 3:
Multiply E = \begin{bmatrix} 2 & 0 \ 1 & 3 \end{bmatrix} (2 × 2) by F = \begin{bmatrix} 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix} (2 × 3).

• Compatibility: Columns of E (2) = Rows of F (2) → Valid.
• Result size: 2 × 3.
• Compute entries:
  • (1,1): (2)(4) + (0)(7) = 8 + 0 = 8
  • (1,2): (2)(5) + (0)(8) = 10 + 0 = 10
  • (1,3): (2)(6) + (0)(9) = 12 + 0 = 12
  • (2,1): (1)(4) + (3)(7) = 4 + 21 = 25
  • (2,2): (1)(5) + (3)(8) = 5 + 24 = 29
  • (2,3): (1)(6) + (3)(9) = 6 + 27 = 33
• Result:
  [ EF = \begin{bmatrix} 8 & 10 & 12 \ 25 & 29 & 33 \end{bmatrix} ]

Example 4 (Non-Commutative):
Let G = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} and H = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}.

• GH:
  • (1,1): (1)(0) + (2)(1) = 0 + 2 = 2
  • (1,2): (1)(1) + (2)(0) = 1 + 0 = 1
  • (2,1): (3)(0) + (4)(1) = 0 + 4 = 4
  • (2,2): (3)(1) + (4)(0) = 3 + 0 = 3
    Result: \begin{bmatrix} 2 & 1 \ 4 & 3 \end{bmatrix}

• HG: