ClassX Video Lessons Subjects Algebra Multiplying a matrix by a column vector | Matrices | Precalculus
Matrix multiplication might seem tricky at first, but once you get the hang of it, it’s not so bad. In this guide, we’ll learn how to multiply a matrix by a vector using a simple example to show you the steps.
We need to multiply matrix A by vector w. This operation is written as Aw.
A =
| 0 | 3 | 5 |
| 5 | 5 | 2 |
This is a 2 by 3 matrix, which means it has 2 rows and 3 columns.
w =
| 3 |
| 4 |
| 3 |
This is a column vector, or you can think of it as a 3 by 1 matrix.
Before we start multiplying, we need to make sure it’s possible. For matrix multiplication to work, the number of columns in the first matrix must match the number of rows in the second matrix.
In our example:
Since these numbers match, we can go ahead with the multiplication.
When you multiply a 2 by 3 matrix by a 3 by 1 vector, you get a 2 by 1 matrix (or a column vector). Let’s calculate each part of the resulting vector step by step.
To find the first entry of the resulting vector, we take the dot product of the first row of matrix A with vector w:
(0 × 3) + (3 × 4) + (5 × 3)
Calculating this gives:
0 + 12 + 15 = 27
Next, we find the second entry by taking the dot product of the second row of matrix A with vector w:
(5 × 3) + (5 × 4) + (2 × 3)
Calculating this gives:
15 + 20 + 6 = 41
Putting both entries together, the result of multiplying matrix A by vector w is:
| 27 |
| 41 |
So, the final output is a column vector with the numbers 27 and 41.
Matrix multiplication, especially with vectors, is a step-by-step process that becomes easier with practice. By breaking it down into simple steps, you can confidently perform these operations and get the right answers.
Form teams and create a relay race where each team member is responsible for calculating a part of the matrix-vector multiplication. The first member calculates the dot product for the first entry, the second member for the second entry, and so on. The team that completes the multiplication correctly and fastest wins!
Use an online interactive tool to manipulate matrices and vectors. Experiment with different sizes and values to see how the resulting vector changes. Try to predict the result before performing the multiplication to test your understanding.
Create a visual art piece using matrices. Assign colors to different numerical values and multiply matrices to generate patterns. Present your artwork to the class and explain the mathematical process behind it.
Research a real-world application of matrix multiplication, such as computer graphics or data analysis. Present your findings to the class, explaining how matrix multiplication is used in that field and why it is important.
Create a storyboard or comic strip that illustrates the step-by-step process of matrix-vector multiplication. Use characters or objects to represent the matrix and vector, and show how they interact to produce the final result.
Matrix – A rectangular array of numbers arranged in rows and columns used to represent linear transformations and solve systems of linear equations. – The matrix representing the transformation was used to rotate the geometric figure in the coordinate plane.
Multiplication – An arithmetic operation that combines two numbers to produce a product, often used in algebra to simplify expressions and solve equations. – The multiplication of two matrices involves summing the products of corresponding entries from the rows and columns.
Vector – A quantity defined by both magnitude and direction, often represented as an ordered list of numbers that can be used in operations such as addition and scalar multiplication. – The vector was used to determine the direction and magnitude of the force applied to the object.
Rows – Horizontal lines of elements in a matrix, each containing a sequence of numbers or variables. – The matrix had three rows and two columns, making it a 3×2 matrix.
Columns – Vertical lines of elements in a matrix, each containing a sequence of numbers or variables. – To find the determinant of a matrix, you must consider both its rows and columns.
Entry – An individual number or element within a matrix, located at the intersection of a specific row and column. – The entry in the second row and third column of the matrix was crucial for solving the equation.
Product – The result of multiplying two or more numbers or expressions, often used in algebra to describe the outcome of multiplication operations. – The product of the two matrices was calculated to find the solution to the system of equations.
Operation – A mathematical process such as addition, subtraction, multiplication, or division applied to numbers or expressions. – The operation of matrix addition involves adding corresponding entries from each matrix.
Result – The final outcome or solution obtained after performing a mathematical operation or series of operations. – After performing the calculations, the result showed that the system of equations had no solution.
Calculation – The process of using mathematical methods to find an answer or solve a problem. – The calculation of the determinant was necessary to determine if the matrix was invertible.
