How To Multiply A 3x3 Matrix By A 3x1 Matrix

How to Multiply a 3x3 Matrix by a 3x1 Matrix: A Comprehensive Guide

Matrix multiplication is a fundamental operation in linear algebra with widespread applications in various fields, including computer graphics, machine learning, physics, and engineering. Understanding how to perform matrix multiplication, particularly multiplying a 3x3 matrix by a 3x1 matrix, is crucial for mastering these areas. This comprehensive guide will walk you through the process step-by-step, explaining the underlying concepts and providing practical examples to solidify your understanding.

Understanding Matrix Dimensions and Multiplication Rules

Before diving into the specifics of multiplying a 3x3 matrix by a 3x1 matrix, let's review the fundamental rules of matrix multiplication. The key concept is compatibility: you can only multiply two matrices if the number of columns in the first matrix equals the number of rows in the second matrix.

• 3x3 Matrix: This matrix has three rows and three columns. We can represent it as:

  [ a b c ]
  [ d e f ]
  [ g h i ]
  

• 3x1 Matrix: This matrix has three rows and one column (also known as a column vector). We can represent it as:

  [ j ]
  [ k ]
  [ l ]
  

To multiply a 3x3 matrix by a 3x1 matrix, the number of columns in the 3x3 matrix (3) must equal the number of rows in the 3x1 matrix (3). Since this condition is met, the multiplication is possible. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. In this case, the resulting matrix will be a 3x1 matrix.

Step-by-Step Matrix Multiplication

The multiplication process involves taking the dot product of each row of the 3x3 matrix with the 3x1 matrix. Let's break down the calculation for each element of the resulting 3x1 matrix:

Step 1: Calculating the first element of the resulting matrix:

To find the first element of the resulting 3x1 matrix, we take the dot product of the first row of the 3x3 matrix and the 3x1 matrix:

(a, b, c) • (j, k, l) = aj + bk + cl

This means we multiply corresponding elements and then sum the results. This becomes the first element in our resulting matrix.

Step 2: Calculating the second element of the resulting matrix:

Next, we repeat the process for the second row of the 3x3 matrix:

(d, e, f) • (j, k, l) = dj + ek + fl

This result forms the second element of our 3x1 matrix.

Step 3: Calculating the third element of the resulting matrix:

Finally, we perform the dot product for the third row of the 3x3 matrix:

(g, h, i) • (j, k, l) = gj + hk + il

This value becomes the third element of our resulting 3x1 matrix.

Putting it all together: The Resulting 3x1 Matrix

After completing these three steps, we have all the elements for our resulting 3x1 matrix:

[ aj + bk + cl ]
[ dj + ek + fl ]
[ gj + hk + il ]

This 3x1 matrix is the product of the 3x3 matrix and the 3x1 matrix.

Illustrative Example

Let's solidify our understanding with a numerical example. Consider the following matrices:

3x3 Matrix (A):

[ 1  2  3 ]
[ 4  5  6 ]
[ 7  8  9 ]

3x1 Matrix (B):

[ 10 ]
[ 11 ]
[ 12 ]

Now, let's multiply matrix A by matrix B:

Step 1:

(1, 2, 3) • (10, 11, 12) = (1 * 10) + (2 * 11) + (3 * 12) = 10 + 22 + 36 = 68

Step 2:

(4, 5, 6) • (10, 11, 12) = (4 * 10) + (5 * 11) + (6 * 12) = 40 + 55 + 72 = 167

Step 3:

(7, 8, 9) • (10, 11, 12) = (7 * 10) + (8 * 11) + (9 * 12) = 70 + 88 + 108 = 266

Therefore, the resulting 3x1 matrix is:

[ 68 ]
[ 167]
[ 266]

Applications of 3x3 and 3x1 Matrix Multiplication

The multiplication of a 3x3 matrix by a 3x1 matrix has numerous applications in various fields. Here are a few examples:

• Transformations in Computer Graphics: 3x3 matrices are often used to represent transformations like rotation, scaling, and shearing in 2D or 3D graphics. The 3x1 matrix represents a point or vector. Multiplying the transformation matrix by the point vector applies the transformation to that point.

• Linear Systems of Equations: A system of three linear equations with three unknowns can be represented in matrix form as a 3x3 matrix multiplied by a 3x1 matrix of unknowns, resulting in a 3x1 matrix of constants. Solving this system involves finding the inverse of the 3x3 matrix.

• Physics and Engineering: Many physical phenomena, such as rotations in mechanics and transformations in electromagnetism, can be described using matrix representations. Multiplying matrices allows for the computation of transformed states or properties.

Beyond the Basics: Further Exploration

While this guide focuses on the fundamental process of multiplying a 3x3 matrix by a 3x1 matrix, the concepts can be extended to larger matrices. Understanding the principles of matrix multiplication is essential for tackling more complex linear algebra problems. Here are some areas to explore further:

• Matrix Transpose: Learning about matrix transposes will help you understand how to multiply matrices in different orders.

• Inverse Matrices: Understanding inverse matrices is crucial for solving systems of linear equations.

• Eigenvalues and Eigenvectors: These concepts are fundamental to understanding the properties and behavior of matrices.

• Determinants and Traces: These properties provide valuable information about matrices and their behavior.

Conclusion

Mastering matrix multiplication, particularly multiplying a 3x3 matrix by a 3x1 matrix, is a cornerstone of linear algebra. Understanding the step-by-step process, the rules of matrix compatibility, and the practical applications of this operation will empower you to tackle complex problems in various fields. By continuing your exploration of linear algebra, you'll unlock powerful tools for solving problems and modeling complex systems. Remember to practice consistently and work through various examples to fully grasp the concepts and solidify your skills. The more you practice, the more intuitive and effortless matrix multiplication will become.