Step-by-Step Matrix Multiplication (2x2, 3x3) Explained

Matrix multiplication is a fundamental operation in linear algebra, widely used in computer graphics, physics, engineering, and data science. Unlike regular number multiplication, matrix multiplication has specific rules and is not commutative (meaning $A \times B$ is generally not equal to $B \times A$).

This guide will break down the process step-by-step, covering both 2x2 and 3x3 matrices.

When Can You Multiply Matrices? (Dimension Compatibility)

Before you can multiply two matrices, say matrix $A$ and matrix $B$ (to get $AB$), their dimensions must be compatible.

If matrix $A$ has dimensions $(m \times n)$ (read as "m rows by n columns") and matrix $B$ has dimensions $(n \times p)$, then:

1. The number of columns in the first matrix ($A$) must be equal to the number of rows in the second matrix ($B$).
2. The resulting product matrix $AB$ will have dimensions $(m \times p)$.

In simple terms: $(m \times \underline{n}) \times (\underline{n} \times p) = (m \times p)$.

Example:

• If $A$ is $2 \times 3$ and $B$ is $3 \times 4$, then $AB$ is $2 \times 4$.
• If $A$ is $2 \times 2$ and $B$ is $3 \times 2$, you cannot multiply $AB$ because the inner dimensions ($2$ and $3$) do not match.

How to Multiply Matrices: The "Row by Column" Rule

To find an element in the product matrix $C = AB$, you take the dot product of a row from the first matrix ($A$) and a column from the second matrix ($B$).

Specifically, the element in row $i$ and column $j$ of the product matrix $C$ (denoted as $c_{ij}$) is found by:

1. Taking the $i$-th row of matrix $A$.
2. Taking the $j$-th column of matrix $B$.
3. Multiplying the corresponding elements from the row and the column.
4. Summing these products.

Let's illustrate with examples.

Example 1: 2x2 Matrix Multiplication

Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $B = \begin{pmatrix} e & f \\ g & h \end{pmatrix}$. The product $C = AB$ will be a $2 \times 2$ matrix.

Let's use specific numbers: $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$

We want to find $C = \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix}$.

Step 1: Calculate $c_{11}$ (Row 1 of A $\times$ Column 1 of B) $c_{11} = (1 \times 5) + (2 \times 7) = 5 + 14 = 19$

Step 2: Calculate $c_{12}$ (Row 1 of A $\times$ Column 2 of B) $c_{12} = (1 \times 6) + (2 \times 8) = 6 + 16 = 22$

Step 3: Calculate $c_{21}$ (Row 2 of A $\times$ Column 1 of B) $c_{21} = (3 \times 5) + (4 \times 7) = 15 + 28 = 43$

Step 4: Calculate $c_{22}$ (Row 2 of A $\times$ Column 2 of B) $c_{22} = (3 \times 6) + (4 \times 8) = 18 + 32 = 50$

So, the product matrix $C$ is: $C = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$

Example 2: 3x3 Matrix Multiplication

Let $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ and $B = \begin{pmatrix} 9 & 8 & 7 \\ 6 & 5 & 4 \\ 3 & 2 & 1 \end{pmatrix}$. The product $C = AB$ will be a $3 \times 3$ matrix.

We need to calculate 9 elements for $C$. Let's go through a few.

Step 1: Calculate $c_{11}$ (Row 1 of A $\times$ Column 1 of B) $c_{11} = (1 \times 9) + (2 \times 6) + (3 \times 3) = 9 + 12 + 9 = 30$

Step 2: Calculate $c_{12}$ (Row 1 of A $\times$ Column 2 of B) $c_{12} = (1 \times 8) + (2 \times 5) + (3 \times 2) = 8 + 10 + 6 = 24$

Step 3: Calculate $c_{13}$ (Row 1 of A $\times$ Column 3 of B) $c_{13} = (1 \times 7) + (2 \times 4) + (3 \times 1) = 7 + 8 + 3 = 18$

...and so on for all 9 elements.

Let’s calculate one element from the second row to make the pattern clear.

Step 4: Calculate $c_{21}$ (Row 2 of A $\times$ Column 1 of B)

$$c_{21} = (4 \times 9) + (5 \times 6) + (6 \times 3) = 36 + 30 + 18 = 84$$

Using the same row-by-column process, we compute the remaining elements:

• $c_{22} = (4 \times 8) + (5 \times 5) + (6 \times 2) = 32 + 25 + 12 = 69$
• $c_{23} = (4 \times 7) + (5 \times 4) + (6 \times 1) = 28 + 20 + 6 = 54$
• $c_{31} = (7 \times 9) + (8 \times 6) + (9 \times 3) = 63 + 48 + 27 = 138$
• $c_{32} = (7 \times 8) + (8 \times 5) + (9 \times 2) = 56 + 40 + 18 = 114$
• $c_{33} = (7 \times 7) + (8 \times 4) + (9 \times 1) = 49 + 32 + 9 = 90$

After calculating all elements, the product matrix $C$ is:

$$C = \begin{pmatrix} 30 & 24 & 18 \\ 84 & 69 & 54 \\ 138 & 114 & 90 \end{pmatrix}$$

Properties of Matrix Multiplication

Matrix multiplication follows several important rules that are useful both theoretically and in practice.

• Not Commutative:
  In general,

  $$AB \neq BA$$

  Even when both products are defined, they often produce different results. In some cases, one product may exist while the other does not due to incompatible dimensions.

• Associative:

  $$(AB)C = A(BC)$$

  This means that when multiplying three matrices, the grouping does not affect the final result (as long as all products are defined).

• Distributive:

  $$A(B + C) = AB + AC$$ $$(A + B)C = AC + BC$$

  Matrix multiplication distributes over matrix addition.

• Identity Matrix:
  There exists an identity matrix $I$ such that:

  $$AI = IA = A$$

  For an $n \times n$ matrix, the identity matrix is an $n \times n$ matrix with ones on the main diagonal and zeros elsewhere.

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Common Mistakes to Avoid

Matrix multiplication errors are very common, especially for beginners. Watch out for these pitfalls:

• Ignoring Dimensions:
  Always check that the number of columns in the first matrix equals the number of rows in the second matrix.

• Element-Wise Multiplication:
  Matrix multiplication is not done by multiplying corresponding elements directly. That operation is called the Hadamard product and is different from standard matrix multiplication.

• Forgetting Order Matters:
  Even if both $AB$ and $BA$ exist, they are usually not equal.

• Arithmetic Errors:
  Because each entry involves multiple multiplications and additions, small arithmetic mistakes can easily occur. Double-check your calculations.

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Why Matrix Multiplication Matters

Matrix multiplication is more than a classroom exercise. It appears in many real-world applications:

• Computer Graphics: Transforming and rotating objects in 2D and 3D space.
• Physics: Describing systems of equations and physical transformations.
• Engineering: Modeling networks, signals, and control systems.
• Data Science & Machine Learning: Linear transformations, neural networks, and optimization algorithms.

Understanding how matrix multiplication works gives you access to powerful tools across science and technology.

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Key Takeaways

• Matrix multiplication follows the row-by-column rule.
• Dimensions must be compatible: $(m \times n)(n \times p) = (m \times p)$.
• The operation is not commutative, but it is associative and distributive.
• Careful, step-by-step calculation prevents most mistakes.
• With practice, matrix multiplication becomes mechanical and intuitive.

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Conclusion

Matrix multiplication may look intimidating at first, but it becomes straightforward once you understand the pattern. By carefully applying the row-by-column rule and checking dimensions, you can confidently multiply both $2 \times 2$ and $3 \times 3$ matrices.

Mastering this operation is a crucial step in linear algebra and opens the door to many advanced topics and real-world applications.