In the world of mathematics, matrices are fundamental tools, especially when dealing with linear transformations and systems of equations. A matrix is essentially an array of numbers arranged in rows and columns. Understanding how to manipulate matrices, particularly matrix multiplication, is crucial. This guide will break down the process of matrix multiplication in a clear, step-by-step manner, making it accessible to anyone eager to learn.
First, let’s get acquainted with the basic structure of a matrix.
Understanding Matrices: Rows, Columns, and Dimensions
A matrix is defined by its number of rows and columns. Rows are the horizontal lines of numbers, and columns are the vertical lines. The dimensions of a matrix are always written as (rows × columns). For instance, the matrix below has 2 rows and 3 columns, making it a 2×3 matrix.
Before diving into multiplying matrices together, let’s start with a simpler operation: scalar multiplication.
Scalar Multiplication: Multiplying a Matrix by a Number
Scalar multiplication involves multiplying a matrix by a single number, known as a scalar. This is a straightforward process: you simply multiply each element in the matrix by the scalar.
For example, if we multiply the matrix by the scalar 2, the calculation for each element is as follows:
| Calculation | Result |
|---|---|
| 2 × 4 = 8 | 8 |
| 2 × 0 = 0 | 0 |
| 2 × 1 = 2 | 2 |
| 2 × -9 = -18 | -18 |
This results in a new matrix where every element is scaled by the factor of 2. Scalar multiplication is a basic but essential operation to understand before moving on to matrix-matrix multiplication.
Matrix Multiplication: Multiplying Matrices Together
Multiplying a matrix by another matrix is a bit more involved than scalar multiplication. It requires understanding the concept of the “dot product” of rows and columns. Let’s illustrate this with an example.
To calculate the element in the first row and first column of the resulting matrix, we take the dot product of the first row of the first matrix and the first column of the second matrix.
The dot product is calculated by multiplying corresponding elements from the row and column and then summing up these products.
For the 1st row and 1st column:
(1, 2, 3) • (7, 9, 11) = (1×7) + (2×9) + (3×11) = 7 + 18 + 33 = 58
This 58 becomes the element in the first row and first column of the result.
Let’s look at another example for the 1st row and 2nd column:
To find the element in the first row and second column of the result, we take the dot product of the first row of the first matrix and the second column of the second matrix:
(1, 2, 3) • (8, 10, 12) = (1×8) + (2×10) + (3×12) = 8 + 20 + 36 = 64
Similarly, for the 2nd row and 1st column:
(4, 5, 6) • (7, 9, 11) = (4×7) + (5×9) + (6×11) = 28 + 45 + 66 = 139
And for the 2nd row and 2nd column:
(4, 5, 6) • (8, 10, 12) = (4×8) + (5×10) + (6×12) = 32 + 50 + 72 = 154
Combining these results, we get the final product matrix:
This method of matrix multiplication might seem complicated at first, but it’s designed to be incredibly useful in various applications. Let’s explore a real-world example to understand why matrix multiplication is defined this way.
Why Matrix Multiplication Works This Way: Real-World Example
Imagine a local shop that sells three types of pies: apple, cherry, and blueberry.
- Apple pies cost $3 each.
- Cherry pies cost $4 each.
- Blueberry pies cost $2 each.
Let’s also say we know how many of each type of pie were sold over four days:
To calculate the total sales value for Monday, we need to multiply the price of each pie type by the number sold and then sum these values.
Sales for Monday = (Price of Apple Pie × Apple Pies Sold) + (Price of Cherry Pie × Cherry Pies Sold) + (Price of Blueberry Pie × Blueberry Pies Sold)
Sales for Monday = ($3 × 13) + ($4 × 8) + ($2 × 6) = $39 + $32 + $12 = $83
This calculation is precisely the “dot product” of the price matrix and the quantity matrix for Monday:
(Price per pie) • (Pies sold on Monday) = ($3, $4, $2) • (13, 8, 6) = $83
We do this for each day to find the total sales for each day:
- Tuesday’s sales: ($3 × 9) + ($4 × 7) + ($2 × 4) = $63
- Wednesday’s sales: ($3 × 7) + ($4 × 4) + ($2 × 0) = $37
- Thursday’s sales: ($3 × 15) + ($4 × 6) + ($2 × 3) = $75
Putting these daily sales values into a matrix form, we get:
This example clearly illustrates why matrix multiplication uses the dot product. It allows us to perform operations that aggregate values in a meaningful way, such as calculating total sales from prices and quantities.
Rules for Matrix Multiplication: Dimensions Matter
When multiplying matrices, the dimensions are crucial. Not all matrices can be multiplied together. The fundamental rule is:
- For matrix multiplication of two matrices, say Matrix A and Matrix B, the number of columns in Matrix A must be equal to the number of rows in Matrix B.
If Matrix A is an m×n matrix and Matrix B is an n×p matrix, then their product, Matrix C = A × B, is an m×p matrix. Notice that the ‘inner’ dimensions (n and n) must match, and the ‘outer’ dimensions (m and p) determine the dimensions of the resulting matrix.
Let’s see this in action: multiplying a 1×3 matrix by a 3×1 matrix results in a 1×1 matrix (a scalar).
[1 2 3] × [4] = [1×4 + 2×5 + 3×6] = [32]
[5]
[6]However, if we multiply a 3×1 matrix by a 1×3 matrix, we get a 3×3 matrix:
[4] × [1 2 3] = [4×1 4×2 4×3] = [4 8 12]
[5] [5×1 5×2 5×3] [5 10 15]
[6] [6×1 6×2 6×3] [6 12 18]Understanding these dimension rules is essential to correctly perform matrix multiplication and interpret the results.
Identity Matrix: The Multiplicative Identity
In the realm of matrices, the “Identity Matrix” plays a role similar to the number “1” in regular multiplication. An identity matrix is a square matrix (same number of rows and columns) with 1s on the main diagonal (from the top-left corner to the bottom-right corner) and 0s everywhere else. It’s denoted by the symbol I.
A key property of the identity matrix is that when you multiply any matrix by the identity matrix (in either order, provided the dimensions allow), the original matrix remains unchanged:
A × I = A
I × A = A
The identity matrix is invaluable in linear algebra, especially in solving systems of linear equations and in matrix inversions.
Order Matters: Matrix Multiplication is Not Commutative
One crucial difference between matrix multiplication and regular number multiplication is that matrix multiplication is generally not commutative. This means that the order in which you multiply matrices matters:
AB ≠ BA (in most cases)
Changing the order of multiplication usually leads to a different result, or might even be undefined due to dimension mismatch.
Let’s consider an example:
Matrix A = [1 2] Matrix B = [2 0]
[3 4] [1 2]
A × B = [ (1×2 + 2×1) (1×0 + 2×2) ] = [ 4 4 ]
[ (3×2 + 4×1) (3×0 + 4×2) ] [ 10 8 ]
B × A = [ (2×1 + 0×3) (2×2 + 0×4) ] = [ 2 4 ]
[ (1×1 + 2×3) (1×2 + 2×4) ] [ 7 10]As you can see, A × B and B × A are different matrices. While there are special cases where AB = BA (like when one of the matrices is an identity matrix or a scalar matrix), in general, matrix multiplication is not commutative.
Conclusion
Matrix multiplication is a fundamental operation in linear algebra with wide-ranging applications in computer graphics, physics, engineering, and data science. While it might seem complex initially, understanding the concept of dot product and the rules of matrix dimensions will make it much more manageable. Remember that matrix multiplication is not just a mathematical abstraction; it’s a powerful tool that helps solve real-world problems, from calculating sales to transforming data. By mastering matrix multiplication, you unlock a deeper understanding of linear algebra and its applications.
