Matrix Multiplication – Method, Definition With Examples

At Brighterly, we strive to make math enjoyable and accessible to children. Today, we’re diving into a fascinating and fundamental mathematical concept: matrix multiplication. Matrices might seem intimidating initially, but they are actually quite interesting and profoundly useful in various fields such as computer graphics, physics, and artificial intelligence.

Matrix multiplication is not merely an operation; it’s a gateway to understanding complex mathematical systems. In this comprehensive guide, we’ll explain what matrix multiplication is, define it, provide methods, and discuss properties of matrix multiplication. We’ll also highlight the differences between scalar and matrix multiplication, and delve into the importance of matrix multiplication in equations. And of course, we’ll present some practice problems to test your newfound skills.

What is Matrix Multiplication?

Matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Unlike the standard multiplication of numbers, matrix multiplication is not commutative. It is a way to combine two matrices to generate a new one that may have completely different properties. Matrix multiplication plays a fundamental role in many areas of mathematics, including algebra, geometry, and computer science.

While you may be used to seeing multiplication as straightforward (like 2×3 = 6), the multiplication of matrices is more complex and requires a different approach. So, let’s start with the definition.

Definition of Matrix Multiplication

Matrix multiplication involves the multiplication of the rows of the first matrix by the columns of the second matrix. In essence, the (i, j) element of the result is obtained by summing the product of the corresponding elements of the i-th row of the first matrix and the j-th column of the second matrix.

In mathematical terms, if A = [aᵢⱼ] and B = [bⱼk] are two matrices, then their product AB is the matrix C = [cᵢk], where cᵢk = Σ aᵢⱼbⱼk (sum over all valid values of j).

Matrix Multiplication Method

Here’s how you perform matrix multiplication, step by step:

  1. Confirm that the number of columns in the first matrix is the same as the number of rows in the second matrix. If not, the matrices cannot be multiplied.
  2. Start from the first row of the first matrix and the first column of the second matrix. Multiply each element of the row by the corresponding element of the column and add all these products.
  3. This sum is the element in the first row and first column of the product matrix.
  4. Repeat this process for each row of the first matrix and each column of the second matrix until all elements of the product matrix are filled.

Properties of Matrix Multiplication

Matrix multiplication has several interesting properties, some of which may seem familiar from your experience with numbers, but others might be quite surprising:

  • Associativity: (AB)C = A(BC)
  • Distributivity: A(B + C) = AB + AC and (A + B)C = AC + BC
  • Not Commutative: In general, AB ≠ BA
  • Zero multiplication: A0 = 0 and 0A = 0
  • Identity Element: For any matrix A, IA = A and AI = A where I is the identity matrix of the appropriate size.

Properties of Square Matrices

Square matrices are matrices that have the same number of rows and columns. These have additional properties beyond the ones mentioned earlier:

  • Main Diagonal: In a square matrix, the main diagonal is the set of elements aᵢⱼ where i = j. This plays a special role in many operations.
  • Determinant: Every square matrix has a determinant, a special number that can be calculated from its elements and has many important properties.
  • Inverse: If the determinant is not zero, the matrix has an inverse. The product of a matrix and its inverse is the identity matrix.

Properties of Rectangular Matrices

Rectangular matrices, which have more rows than columns or vice versa, also have special properties. However, many operations such as the determinant or the inverse do not apply to them. Yet, they are still crucial in many areas of mathematics and computer science, including systems of linear equations, transformations in graphics, and machine learning algorithms.

Differences Between Matrix Multiplication and Scalar Multiplication

Matrix multiplication is a more complex process than scalar multiplication. Scalar multiplication involves just multiplying every element of the matrix by a scalar value, whereas matrix multiplication involves a methodical process of taking dot products of rows and columns.

In scalar multiplication, the order of multiplication doesn’t matter, whereas in matrix multiplication, it does matter. Also, while you can always multiply any two numbers, you can only multiply two matrices if their dimensions are compatible.

Equations involving Matrix Multiplication

Matrix multiplication is often used in the formulation of various equations. These equations are used in a variety of fields, from physics to computer graphics to machine learning.

Writing Equations with Matrix Multiplication

When writing equations involving matrix multiplication, the key is to remember the rules and properties of matrix multiplication. For example, if A, B, and C are matrices, and B and C can be multiplied, then you could write an equation like A = BC.

Solving Equations with Matrix Multiplication

Solving equations involving matrix multiplication often involves finding the inverse of a matrix. The inverse of a matrix A is a matrix B such that AB = BA = I, where I is the identity matrix. When such a matrix exists, we can use it to solve equations of the form AX = B by multiplying both sides by A⁻¹ to get X = A⁻¹B.

Practice Problems on Matrix Multiplication

To better understand matrix multiplication, let’s look at some concrete examples. Here are some practice problems to provide you with hands-on experience in performing matrix multiplication and solving related problems.

  1. Multiply the following matrices:

    A = [[2, 3], [4, 1]]
    B = [[1, 2], [5, 6]]

  2. Compute the product of the given matrices:

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

  3. If A = [[2, 4], [1, 1]] and B = [[6, 9], [5, 2]], find the matrices AB and BA. Are they equal?

  4. Given the 3×3 matrix A = [[3, 2, 1], [1, 2, 3], [2, 3, 1]], compute the square of A (AA or A²).

  5. Suppose we have the matrices A = [[1, 3], [2, 0]] and B = [[1, 2], [3, 4]]. Solve the matrix equation AX = B for X.

After trying these problems, you can compare your solutions with the answers provided. If you encounter any difficulties, don’t hesitate to review the method and properties of matrix multiplication discussed above.

Conclusion

Matrix multiplication is a profound concept with far-reaching applications. As we have seen, it isn’t just an abstract idea, but a powerful tool that enables us to solve complex problems in mathematics, physics, computer science, and many other fields. At Brighterly, we believe that understanding such concepts from a young age can set the foundation for a lifetime of curiosity and learning.

Remember, practice is key to mastering matrix multiplication, so be sure to tackle the practice problems we’ve provided. Once you’ve mastered this concept, you’ll find that you’re able to understand and solve more complex mathematical problems.

Frequently Asked Questions on Matrix Multiplication

What is Matrix Multiplication?

Matrix multiplication is an operation where we take two matrices and produce a new matrix. Unlike ordinary multiplication, matrix multiplication involves a methodical process of taking dot products of rows and columns of the involved matrices.

How is Matrix Multiplication Defined?

Matrix multiplication is defined as a binary operation that takes a pair of matrices and produces another matrix. For matrices A and B, the element in the i-th row and j-th column of the resulting matrix is the sum of the product of elements from the i-th row of A and the j-th column of B.

How do You Multiply Matrices?

To multiply matrices, start with the first row of the first matrix and the first column of the second. Multiply each element of the row with the corresponding element of the column and add all the products. This sum becomes the element in the first row and first column of the product matrix. This process is repeated for each row of the first matrix and each column of the second matrix until all elements of the product matrix are filled.

What are the Properties of Matrix Multiplication?

Matrix multiplication is associative and distributive, but not commutative. This means that the order in which matrices are multiplied can change the result. There’s also an identity matrix, which when multiplied with any matrix, doesn’t change the original matrix.

What are Square and Rectangular Matrices?

Square matrices have the same number of rows and columns. They have additional properties and can be diagonal, symmetric, or even identity matrices. Rectangular matrices, on the other hand, have more rows than columns, or vice versa.

What is the Difference Between Matrix and Scalar Multiplication?

Scalar multiplication involves multiplying every element of the matrix by a scalar value, while matrix multiplication involves a process of taking dot products of rows and columns. The order of multiplication doesn’t matter in scalar multiplication, while it does in matrix multiplication.

How to Write and Solve Equations Using Matrix Multiplication?

Equations using matrix multiplication are written following the rules of matrix multiplication. To solve such equations, you often need to find the inverse of a matrix. The inverse of a matrix A is a matrix B such that AB = BA = I, where I is the identity matrix. When such a matrix exists, you can solve equations of the form AX = B by multiplying both sides by A⁻¹ to get X = A⁻¹B.

Information Sources:
  1. Wikipedia – Matrix multiplication
  2. Wolfram MathWorld – Matrix Multiplication
  3. MIT OpenCourseWare – Matrix Multiplication

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