Table of Contents

Matrix multiplication

## Introduction to Matrix multiplication

A fundamental operation in linear algebra called matrix multiplication involves multiplying two matrices to create a new matrix. It enables many applications in disciplines including mathematics, computer science, physics, and engineering by facilitating the systematic modification and combining of data..

### Definition of Matrix multiplication

A new matrix is created by multiplying two existing ones using a binary process called matrix multiplication. It is defined for matrices when the first matrix’s column’s number equals the second matrix’s row number.

Given two matrices, say matrix A with m rows and n columns and matrix B with n rows and p columns, the resulting matrix C will have a dimension of m rows and p columns when A and B are multiplied.

In more detail, each element in the i-th row of matrix A is multiplied by the corresponding element in the j-th column of matrix B, and the products are added to obtain the element at the i-th row and j-th column of matrix C.

Where C = A * B, the general formula for matrix multiplication is:

For , k = 1, 2, 3, …, n, c_{ij} = sum( a_{ik} x b_{kj })

Here the element c_{ij} at the i-th row and j-th column of matrix C, the element a_{ik} is at the i-th row and k-th column of matrix A, and the element b_{kj} is at the k-th row and j-th column of matrix B.

### Matrix multiplication by a scalar

The process of multiplying each element of a matrix by a scalar value is known as scalar multiplication. Any real number or complex number can be the scalar. Each component is multiplied by the scalar value, creating a new matrix with the same dimensions as the original matrix.

### Limitations for matrix multiplication

- Only matrices where the first matrix’s column’s number equals the second matrix’s row’s number are those for which matrix multiplication is defined.
- The sequence of multiplication has an impact on the outcome since matrix multiplication is not commutative.
- The size of the input matrices affects the dimensions of the final matrix. The resulting matrix will have an equal number of rows to the first matrix’s rows and an equal number of columns to the second matrix’s columns.
- Given that matrix multiplication necessitates a sizable number of scalar multiplications and additions, it can be computationally expensive, especially for big matrices.
- If you try to multiply a matrix with incompatible dimensions, an error will occur because matrix multiplication is not defined for such matrices.

### Steps to find the multiplication of two matrices

The stages involved in multiplying two matrices, commonly designated as A and B, are as follows:

- Verify whether the matrices may be multiplied: The number of rows in matrix B and the number of columns in matrix A must match.
- The size of the final matrix should be determined. The number of rows from matrix A and the number of columns from matrix B will be present in the final matrix.
- Compute the total of the products of the respective elements from the row in matrix A and the column in matrix B for each element in the resulting matrix C. This is known as element-wise multiplication and summation.
- Once all of the components of the resulting matrix C have been determined, repeat the procedure for each component.
- The combination of matrices A and B yields matrix C.

### Properties of matrix multiplication

- Associative property : For matrices A, B, and C of compatible dimensions, associativity states that (A * B) * C = A * (B * C). It is possible to rearrange the matrix multiplication order without impacting the outcome.
- Distributive property : A * (B + C) = A * B + A * C for matrices A, B, and C with compatible dimensions. Matrix addition is distributed over via matrix multiplication.
- Scalar Multiplication: If the matrices A and B are of compatible dimensions and the scalar value k is used, then (k * A) * B = k * (A * B) = A * (k * B). It is possible to use scalar multiplication either before or after matrix multiplication.
- Identity Matrix: Where I is the identity matrix, for any matrix A of suitable dimensions, A * I = I * A = A. When a matrix is multiplied by the identity matrix, the matrix remains intact.
- A * 0 = 0 * A = 0, where 0 is the zero matrix, for any matrix A of the proper dimensions. The zero matrix is obtained by multiplying a matrix by the zero matrix.
- Non-commutativity: In general, Matrix multiplication does not commutate., A * B does not always equal B * A.
- Dimension compatibility: For matrix multiplication to be defined, the first matrix’s number of columns must match the second matrix’s number of rows.

### Problems on Matrix multiplication

- Linear Function f(x) = 2x + 3. To generate the output, this function takes an input value x, multiplies it by 2, and then adds 3.
- The quadratic function is defined as f(x) = x
^{2}-4x + 5. This function includes the square of the input value x, as well as linear and constant terms. - f(x) = 2
^{x }is an exponential function. To compute the output, this function raises the base (2) to the power of the input value x. - f(x) = sin(x) is a trigonometric function. The sine of the input value x is computed using this function.
- f(x) = x is the identity function. Without any alteration, this function just returns the input value as the output value.

## FAQs on Matrix multiplication

### What is meant by matrix multiplication?

When two matrices are multiplied, a new matrix is produced. This operation is known as matrix multiplication. In order to do this, relevant items from the first matrix's rows and the second matrix's columns must be multiplied by one another and then added. Based on the sizes of the original matrices, the final matrix has the following dimensions.

### How to multiply two given matrices?

Make sure the number of columns in the first matrix and the number of rows in the second matrix match before multiplying the two matrices. Add the matching elements from the rows and columns after multiplying them. To obtain the resulting matrix dimensions, repeat for each element.

### Is it possible to combine any two matrices?

No, the number of columns in the first matrix must match the number of rows in the second matrix in order for matrix multiplication to be defined.

### Is commutative matrix multiplication possible?

The multiplication of matrices is not commutative. In most cases, . AB ≠ BA. It concerns what order the matrices are in.

### What role does matrix multiplication play?

Solving systems of linear equations, linear algebra, computer graphics, and other disciplines all depend on matrix multiplication. It makes transformations, compositions, and mathematical operations involving several variables effectively representable and manipulable.

### Using a calculator or software, how do I multiply matrices?

Matrix multiplication functions are available in the majority of calculators and software programmes. Simply enter the matrices, and the programme will multiply them and output the final matrix for you.

### Is multiplying any two matrices always possible?

No, it is only possible to multiply two matrices if the number of columns in the first matrix and the number of rows in the second matrix are equal. The matrices cannot be multiplied in any other case.

### Is it possible for the final matrix to differ in size from the initial matrices?

The final matrix can indeed be of a different size than the underlying matrices. The first matrix establishes the number of rows, and the second matrix establishes the number of columns.