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 just show the full calculation for one more, $c_{21}$:

Step 4: Calculate $c_{21}$ (Row 2 of A $\times$ Column 1 of B) $c_{21} = (4 \times 9) + (5 \times 6) + (6 \times 3) = 36 + 30 + 18 = 84$

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

30 & 24 & 18 \\ 84 & 69 & 54 \\ 138 & 114 & 90 \end{pmatrix}$$ ## Properties of Matrix Multiplication - **Not Commutative:** In general, $AB \neq BA$. You might even be able to compute $AB$ but not $BA$ if dimensions don't match for the reverse order. - **Associative:** $(AB)C = A(BC)$. This means the order of operations for multiple multiplications doesn't matter for grouping. - **Distributive:** $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$. - **Identity Matrix:** There is an identity matrix, $I$, such that $AI = IA = A$. For $n \times n$ matrices, $I$ is an $n \times n$ matrix with ones on the main diagonal and zeros elsewhere. ## Common Mistakes - **Incorrect Dimensions:** Always check if the inner dimensions match before attempting to multiply. - **Element-wise Multiplication:** Do not multiply matrices by simply multiplying corresponding elements (that's a different operation called the Hadamard product). - **Order Matters:** Remember that $AB \neq BA$ in most cases. - **Calculation Errors:** Matrix multiplication involves a lot of arithmetic. Double-check your additions and multiplications! ## Conclusion Matrix multiplication, while initially seeming complex, becomes straightforward with practice. The key is to consistently apply the "row by column" rule, ensuring your matrix dimensions are compatible. Master this operation, and you'll unlock a powerful tool for solving problems in various scientific and engineering disciplines.