Matrix Multiplication and Its Basic Formula

Matrix multiplication is one of the most important operations in mathematics and is widely used in algebra, physics, computer science, engineering, economics, statistics, and data science. It provides a systematic way to combine information represented by two matrices and produce a new matrix.

At first, matrix multiplication can seem more complicated than ordinary multiplication because the numbers are not multiplied in the same positions. Instead, the entries of one row are multiplied by the entries of a column, and the resulting products are added together. Once this row-and-column process is understood, matrix multiplication becomes much easier to perform.

Before multiplying two matrices, it is also important to check their dimensions. Not every pair of matrices can be multiplied. The number of columns in the first matrix must be equal to the number of rows in the second matrix. This condition determines whether multiplication is possible and also tells us the size of the resulting matrix.

In this article, we will learn what matrix multiplication means, how to check whether two matrices can be multiplied, understand its basic formula, work through examples, and examine some important properties and common mistakes.

What Is Matrix Multiplication?

Matrix multiplication is an operation used to multiply two matrices and obtain a third matrix. The entries of the resulting matrix are calculated by combining the rows of the first matrix with the columns of the second matrix.

Consider two matrices, A and B. If the number of columns in A is equal to the number of rows in B, then the product AB can be calculated.

For example, suppose A has dimensions 2 × 3 and B has dimensions 3 × 2. The multiplication is possible because the number of columns in A is 3 and the number of rows in B is also 3.

The resulting matrix will have dimensions 2 × 2.

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A: 2 × 3
B: 3 × 2
AB: 2 × 2

This gives us the most important dimension rule for matrix multiplication.

The Dimension Rule for Matrix Multiplication

To multiply two matrices, the inner dimensions must be equal.

Suppose:

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A = m × n
B = n × p

Then:

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AB = m × p

The two middle numbers, n and n, must be the same.

For example:

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(2 × 3)(3 × 4) = 2 × 4

Matrix multiplication is possible because the inner dimensions are both 3.

Another example is:

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(3 × 2)(2 × 5) = 3 × 5

Again, the inner dimensions are both 2, so multiplication is possible.

However:

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(2 × 3)(2 × 4)

cannot be multiplied because the first matrix has 3 columns while the second matrix has 2 rows.

Therefore, the multiplication is not defined for these two matrices in this order.

Basic Formula of Matrix Multiplication

The basic formula for matrix multiplication follows the row-by-column rule.

Suppose matrix A has entries aᵢⱼ and matrix B has entries bᵢⱼ. Each entry of the product AB is found by multiplying the elements in a particular row of A by the corresponding elements in a particular column of B and adding the results.

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(AB)ᵢⱼ = aᵢ₁b₁ⱼ + aᵢ₂b₂ⱼ + aᵢ₃b₃ⱼ + ... + aᵢₙbₙⱼ

A shorter general form is:

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(AB)ᵢⱼ = Σ aᵢₖbₖⱼ

Here, the symbol Σ means that the products are added together over the required values of k.

For beginners, however, the row-by-column method is usually easier to understand than the compact summation notation.

How Matrix Multiplication Works

The easiest way to understand matrix multiplication is to look at a simple example.

Consider the following two matrices:

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A = [ 1  2 ]
    [ 3  4 ]
B = [ 5  6 ]
    [ 7  8 ]

Both matrices are 2 × 2, so their multiplication is possible.

To find the first entry of AB, take the first row of A and the first column of B.

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First row of A:       1   2
First column of B:    5   7
1 × 5 + 2 × 7
= 5 + 14
= 19

Therefore, the top-left entry of the product is 19.

Now find the top-right entry by using the first row of A and the second column of B.

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1 × 6 + 2 × 8
= 6 + 16
= 22

The first row of the resulting matrix is therefore 19, 22.

Next, use the second row of A.

For the bottom-left entry:

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3 × 5 + 4 × 7
= 15 + 28
= 43

For the bottom-right entry:

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3 × 6 + 4 × 8
= 18 + 32
= 50

Therefore:

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AB = [ 19  22 ]
     [ 43  50 ]

This example demonstrates the central idea of matrix multiplication: row of the first matrix × column of the second matrix.

Matrix Multiplication Formula for Two 2 × 2 Matrices

For two general 2 × 2 matrices, the multiplication can be written as follows.

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A = [ a  b ]
    [ c  d ]
B = [ e  f ]
    [ g  h ]

The product AB is:

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AB = [ ae + bg    af + bh ]
     [ ce + dg    cf + dh ]

Each entry is formed by multiplying the appropriate row and column elements and then adding the products.

For example, the top-left element is:

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ae + bg

because the first row of A is combined with the first column of B.

The top-right element is:

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af + bh

because the first row of A is combined with the second column of B.

The same process is repeated for the second row.

Matrix Multiplication of a 2 × 3 and 3 × 2 Matrix

Matrix multiplication becomes more interesting when the matrices have different dimensions.

Consider:

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A = [ 1  2  3 ]
    [ 4  5  6 ]
B = [ 7   8 ]
    [ 9  10 ]
    [ 11 12 ]

Matrix A is 2 × 3, while matrix B is 3 × 2.

The multiplication is possible because:

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(2 × 3)(3 × 2)

The inner dimensions are both 3.

The resulting matrix will have dimensions:

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2 × 2

Now calculate each entry.

For the first entry:

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1 × 7 + 2 × 9 + 3 × 11
= 7 + 18 + 33
= 58

For the second entry:

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1 × 8 + 2 × 10 + 3 × 12
= 8 + 20 + 36
= 64

For the third entry:

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4 × 7 + 5 × 9 + 6 × 11
= 28 + 45 + 66
= 139

For the fourth entry:

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4 × 8 + 5 × 10 + 6 × 12
= 32 + 50 + 72
= 154

Therefore:

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AB = [ 58   64 ]
     [ 139  154 ]

Notice that the resulting matrix has two rows and two columns, exactly as predicted by the dimension rule.

Why Rows and Columns Matter

Rows and columns play different roles in matrix multiplication.

The rows of the first matrix determine the rows of the resulting matrix. The columns of the second matrix determine the columns of the resulting matrix.

For example:

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(3 × 4)(4 × 2) = 3 × 2

The first matrix has 3 rows, so the result has 3 rows. The second matrix has 2 columns, so the result has 2 columns.

The four inner dimensions disappear after confirming that multiplication is possible.

This provides a useful way to remember the rule:

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Outer dimensions → dimensions of the answer
(3 × 4)(4 × 2)
3 × 2

Matrix Multiplication Is Not Entry-by-Entry Multiplication

One common mistake is to multiply corresponding entries of two matrices.

For example, if:

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A = [ 1  2 ]
    [ 3  4 ]
B = [ 5  6 ]
    [ 7  8 ]

ordinary entry-by-entry multiplication would produce:

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[ 1×5   2×6 ]
[ 3×7   4×8 ]

But this is not the standard matrix product AB.

The standard matrix product is obtained using row-by-column multiplication:

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AB = [ 19  22 ]
     [ 43  50 ]

Entry-by-entry multiplication is a different operation and should not be confused with ordinary matrix multiplication.

Is Matrix Multiplication Commutative?

For ordinary numbers, multiplication is commutative. For example:

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3 × 5 = 5 × 3

However, matrix multiplication generally does not have this property.

In general:

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AB ≠ BA

There are several reasons for this.

First, AB may be defined while BA is not. Even when both products are defined, they may produce different matrices.

For example, using the matrices above:

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A = [ 1  2 ]
    [ 3  4 ]
B = [ 5  6 ]
    [ 7  8 ]

we found:

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AB = [ 19  22 ]
     [ 43  50 ]

Now calculate BA:

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BA = [ 23  34 ]
     [ 31  46 ]

Clearly:

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AB ≠ BA

Therefore, the order of matrices matters.

Important Properties of Matrix Multiplication

Matrix multiplication follows several useful mathematical properties.

Associative Property

When the dimensions allow all the products to be calculated, matrix multiplication is associative.

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(AB)C = A(BC)

This means that when multiplying three matrices, the grouping can be changed without changing the final result.

Distributive Property

Matrix multiplication is distributive over matrix addition.

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A(B + C) = AB + AC

Similarly:

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(A + B)C = AC + BC

These properties are useful when simplifying matrix expressions.

Identity Matrix

The identity matrix acts like the number 1 in ordinary multiplication.

For a 2 × 2 matrix, the identity matrix is:

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I = [ 1  0 ]
    [ 0  1 ]

For a suitable matrix A:

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AI = IA = A

The identity matrix leaves the original matrix unchanged.

Matrix Multiplication and Linear Transformations

Matrix multiplication is not only a calculation technique. It also has an important geometric meaning.

Matrices can represent transformations such as rotations, reflections, scaling, and other changes to vectors. When two transformation matrices are multiplied, the resulting matrix represents the combined transformation.

For example, one matrix might represent a rotation while another represents a scaling operation. Their product can represent both operations together.

This is one reason matrix multiplication is so important in computer graphics, robotics, physics, engineering, and computer science.

Applications of Matrix Multiplication

Matrix multiplication is used in many practical fields.

Computer Graphics

Computer graphics use matrices to transform objects. Scaling, rotation, translation, and projection can be represented using matrices.

Physics

Matrices are used to represent transformations, quantum mechanical operators, systems of equations, and many other mathematical structures in physics.

Engineering

Engineers use matrices to model systems, solve simultaneous equations, analyze structures, and perform transformations.

Computer Science

Matrix operations are fundamental in algorithms, machine learning, image processing, computer vision, and artificial intelligence.

Data Science and Machine Learning

Large datasets can be represented using matrices. Matrix multiplication is used in neural networks, linear regression, dimensional transformations, and many machine-learning calculations.

Economics

Matrices can represent relationships between different economic sectors, quantities, and variables. Matrix multiplication can then be used to calculate combined effects.

Common Mistakes in Matrix Multiplication

Understanding the most common errors can make matrix multiplication much easier.

1. Ignoring the Dimensions

Always check whether the number of columns in the first matrix equals the number of rows in the second matrix.

2. Multiplying Corresponding Entries

Do not simply multiply entries in the same positions. Standard matrix multiplication uses rows and columns.

3. Using the Wrong Column

When calculating one entry, make sure you select the correct column from the second matrix.

4. Forgetting to Add the Products

Each entry is usually the sum of several products. Multiplying the numbers without adding them will give an incorrect result.

5. Reversing the Order

Remember that AB and BA are generally different.

6. Getting the Resulting Dimensions Wrong

If:

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(m × n)(n × p)

then the answer must be:

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m × p

The outer dimensions determine the size of the product.

A Simple Method to Multiply Matrices

A reliable method can be summarized in a few steps.

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Step 1: Check the dimensions.
Step 2: Confirm that the columns of the first matrix
        equal the rows of the second matrix.
Step 3: Determine the dimensions of the result
        using the outer dimensions.
Step 4: Take one row from the first matrix.
Step 5: Take one column from the second matrix.
Step 6: Multiply corresponding elements.
Step 7: Add the products.
Step 8: Place the result in the appropriate position.
Step 9: Repeat until every entry has been calculated.

Following these steps reduces the chance of mixing up rows and columns.

Summary of Matrix Multiplication

Matrix multiplication is a structured operation based on the interaction between rows and columns. Before multiplying two matrices, their dimensions must be checked carefully. If the first matrix has dimensions m × n and the second has dimensions n × p, their product exists and has dimensions m × p.

The most important formula is based on multiplying the elements of a row in the first matrix by the corresponding elements of a column in the second matrix and adding the products.

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(AB)ᵢⱼ = aᵢ₁b₁ⱼ + aᵢ₂b₂ⱼ + ... + aᵢₙbₙⱼ

The key idea to remember is simple: take a row from the first matrix, take a column from the second matrix, multiply corresponding entries, add the products, and place the result in the correct position.

Matrix multiplication also has important properties such as associativity and distributivity, while it generally does not satisfy the commutative property. Its applications extend far beyond basic mathematics, making it an essential concept in science, engineering, computing, data science, and many other fields.

Once the row-by-column rule becomes familiar, even larger matrix multiplication problems can be approached systematically and accurately.