Matrix math addition

This blog will dive into three fundamental Matrix math Addition, Subtraction, and Multiplication

Matrix Addition

Matrix addition involves combining two matrices by adding their corresponding elements. For this operation to be valid, the matrices must have the same dimentions (i.e., the same number of rows and columns).

How to do matrix addition?

The Formula:

If A= and B= are two matrices, their sum C=A+B is given by:

Example:

The sum C=A+B is computed element-wise:

Key Properties:

• Commutativity: A+B=B+A 
• Associativity: (A+B)+C=A+(B+C)
• Additive Identity: Adding a zero matrix (a matrix with all elements as zero) leaves the original matrix unchanged: A+0=A

Matrix Subtraction

Matrix substitution (or subtraction) is similar to addition except that the components of one matrix are subtracted from the equivalent elements of another. The dimensions of the matrices must be the same, just like in addition.

How to do Matrix Subtraction?

The Formula:

If A= and B= are two matrices, their difference C=A−B is given by:

Example:

The difference C=A−B is:

Key properties:

• Non-Commutative: In general, A−B≠B−A
• Associativity: Subtraction respects the grouping order: (A−B)−C=A−(B+C)
• Subtractive Identity: Subtracting a zero matrix leaves the original matrix unchanged: A−0=A

Matrix Multiplication

Matrix multiplication is a fundamental mathematical operation that involves multiplying two matrices to create a third matrix. Unlike addition, matrix multiplication involves the dot product of rows and columns rather than individual elements.

How to do Matrix Multiplication?

The Formula:

If A=  is an m×n matrix and B= is an n×p matrix, then their product C=A.B= is given by:

This means each element in the resulting matrix $C$ is the dot product of the i-th row of $A$ with the $k$-th column of B.

Example:

The product C=A.B is calculated as follows:

Key properties:

• Non-Commutative: In general, A⋅B≠B⋅A
• Associative: (A⋅B)⋅C=A⋅(B⋅C)
• Distributive: A⋅(B+C)=A⋅B+A⋅C
• Identity Matrix: Multiplying any matrix  A by an identity matrix  I (of appropriate size) does not modify A.

A⋅I=I⋅A=A

Matrix math Addition, Subtraction, and Multiplication are fundamental techniques in linear algebra, creating the groundwork for more complex operations such as inversion, and transformations.Whether you’re just getting started with linear algebra or going deep into applied mathematics, learning these operations can help you gain mathematical fluency and problem-solving confidence.