Last Updated on March 22, 2025 by Maged kamel
A Solved problem for matrix operations.
Matrix operations play a crucial role in various fields, including computer graphics, data science, and engineering. To better understand these operations, let’s explore some real-world applications, such as how image processing relies on matrix transformations.
For instance, in image processing, an image can be represented as a matrix where each element corresponds to a pixel’s color value. Operations like blurring or sharpening an image can be achieved through matrix multiplication.
Review of the commutative, associative, and distributive properties.
Before we proceed to solve a problem for matrix operations, we will review the commutative, associative, and distributive properties, which are foundational concepts in linear algebra. Understanding these properties will help you grasp the significance of matrices in various applications.
What is the commutative property?
To illustrate this further, consider two matrices: A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. When we perform A + B, we get [[6, 8], [10, 12]], and if we switch the order to B + A, we still get [[6, 8], [10, 12]]. This example showcases the commutative property effectively.
The associative property involves grouping numbers and states that when three or more numbers are added or multiplied, the sum or product remains the same regardless of how the numbers are grouped. For example, (A + B) + C = A + (B + C) for matrices.
The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. This property applies to addition and multiplication, but it does not hold for subtraction and division. For example, if we have two matrices A and B, adding them in either order yields the same result: A + B = B + A. This fundamental property of matrices is crucial in simplifying complex algebraic expressions and calculations.
What is the associative property?
This property states that when three or more numbers are added (or multiplied), the sum (or the product) is the same regardless of the grouping of the addends (or the multiplicands).
Grouping means the use of parentheses or brackets to group numbers.
To see this in action, let’s say we have three matrices A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]], and C = [[9, 10], [11, 12]]. Calculating (A + B) + C gives us [[15, 18], [21, 24]], while A + (B + C) also results in [[15, 18], [21, 24]]. This example further demonstrates the associative property.
Associative property involves three or more numbers.
The numbers that are grouped within parenthesis or brackets become one unit.
Associative property can only be used with addition and multiplication, not subtraction or division.
The distributive property, on the other hand, allows us to distribute multiplication over addition, which is invaluable in simplifying expressions. When applying this to matrices, if we have a scalar k and a matrix A, then k(A + B) = kA + kB. This property is essential in solving matrix equations efficiently.
For example, if k = 2 and A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]], we can compute 2(A + B) = 2[[6, 8], [10, 12]] = [[12, 16], [20, 24]], and separately calculate 2A + 2B to confirm that both methods yield the same result.
What is distributive property?
Now that we have a solid understanding of these properties, let’s examine how we can apply them in matrix addition. It’s important to recognize the significance of these properties in computational tasks and mathematical proofs, particularly in linear transformations.
The distributive property is one of the most frequently used properties in math. In general, this term refers to the distributive property of multiplication, which states that the
Definition: The distributive property lets you multiply a sum by multiplying each addend separately and adding the products.
for instance 3*(2+3)=(3*2)+(3*3)=15. The next slide image contains a review of the three properties of Commutative, Associative, and Distributive Properties.
How can we apply these properties in matrix addition?
The first addition is commutative since adding two matrices A and B, is the same as adding B+A and also the multiplication of ab=ba..
Properties of matrix addition.
In our next example, we will tackle a numerical problem that illustrates these properties in action. Matrix operations can often be visualized geometrically, such as how they affect vectors in space. Understanding this aspect can provide deeper insights into the nature of matrices.
As quoted from Prof. Kuldeep Singh’s handbook, five properties of matrix addition are included.
As we continue, it’s beneficial to consider other scenarios where matrix operations are applied, such as in computer graphics for transformations like scaling, rotation, and translation of objects. These operations use matrix multiplication extensively to alter the coordinates of images and shapes on screen.
Furthermore, learning about the implications of matrix addition and the properties involved can provide insights into more advanced topics, such as eigenvalues and eigenvectors, which are pivotal in areas like machine learning and data analysis.
A Solved problem for matrix operations.
Our next post will explore the Transpose of a Matrix, its definitions, and solved problems. This topic is essential for understanding matrix operations in greater depth.
As a practice to Matrix operations-part-2, we will use a solved problem with multiple operations to perform the necessary matrix operations.
Given three matrices, A, B, and C, one must check their properties by estimating A+B and then B+A to prove that A+B=B+A through a numerical solved problem. The A matrix is (1 -1 3 7, 2 9 5 -6), while the B matrix is (7 6 -3 2, 1 4 5 3), and the C matrix is(-2 -7 8 6, 3 -9 2 1). The following slide shows part a) solution, which includes adding A+B.
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The following slide shows part b) solution, which includes adding B+A, similar to part a).
The next task includes parts c and d to estimate (A+B) + C and compare the result with A+(B+C) to prove the associative law. Add three matrices together for parts c and D by using different expressions.
Additional parts are solved, as shown in the next slide image. Adding Matrix A to zero and the value of adding five A matrices is equal to 5A. For part g, Adding C+(-c) will lead to a zero matrix.
For those interested in further explorations of matrices, I recommend visiting Math is Fun. It’s a fantastic resource for understanding different types of matrices and their properties.
This is the next post, Transpose of a Matrix (definitions-solved problem).
