Matrix Operations

Matrices are very informative structures in mathematics and are applicable in most fields of computer science, physics and engineering. The operations involved with matrices are addition, subtraction and multiplication of matrices. A matrix is an arrangement of numbers or expressions in the form of rows and columns in a two-dimensional structure called a rectangular array. This article explores basic matrix operations, including matrix addition and subtraction, scalar multiplication, matrix product, and matrix transpose.

1.0Operations on Matrices

Addition, subtraction and multiplication are the elementary operations on the matrix. In order to add or subtract a matrix, these must be of the same order, whereas for multiplication purposes, the no. of columns in the 1st matrix should be equal to the no. of rows in the second matrix.

1. Addition of Matrices
2. Subtraction of Matrices
3. Scalar Multiplication of Matrices
4. Multiplication of Matrices

Addition and Subtraction

Two matrices can be added or subtracted if they have the same dimensions. The operation is performed element-wise:

Sum of two Matrices:

If A = [ 1  2 ]

[3  4  ]

and B = [5  6 ] 

[ 7  8 ]

Then A  + B =  [1+5  2+6 ] 

[ 3+7  4+8 ]

=   [6       8 ] 

 [ 10  12 ]

Difference of two Matrices:

A–B = [1–5  2–6  6–8]

[3–7    4–8   9–7]

[2–4    5–6   8–5]

= [-4   -4   -2]

[-4    -4   2]

[-2    -1   -3]

Scalar Multiplication

In scalar multiplication, each element of the matrix is multiplied by a constant k:

If k = 3 and A = [1  2]

[3  4]

Then kA = [3 x 1   3 x 2 ] 

[ 3 x 3  3 x 4 ]

=   [3     6 ] 

[ 9  12 ]

Matrix Operation Multiplication

Matrix multiplication is not performed element-wise. Instead, it involves the rows of the first matrix & the columns of the second matrix. Two matrices, A & B, can be multiplied if the no. of columns in A equals the no. of rows in B.

If A = [ 1  2 ]

[3  4  ]

and B = [5  6 ] 

[ 7  8 ]

Then, AB =  [(1x5 + 2x7) (1x6 + 2x8) ] 

[ (3x5 + 4x7) ( 3x6 + 4x8)]

= [19 22 ] 

[ 43 50 ]

Matrix multiplication is widely used in transformations, such as in computer graphics and in solving systems of linear equations.

2.0Inverse Operation of a Matrix

The inverse operation of a matrix is finding the inverse matrix, which is denoted as A⁻¹, such that when multiplied by the original matrix A, the result will be the identity matrix I (where I is a square matrix with ones on the diagonal and zeros elsewhere).

In maths, it can be written as: 

A×A⁻¹ = A⁻¹×A = I

The formula to find the Inverse of a matrix is: 

A-1=1AAdj A

3.0Elementary Matrix Operations

Elementary matrix operations are transformations applied to matrices to simplify them. These include:

• Row Addition/Subtraction: Adding or subtracting rows.
• Row Multiplication: Multiplying a row by a non-zero scalar.
• Row Interchange: Swapping two rows.

These operations are crucial in solving systems of linear equations using Gaussian elimination.

4.0Transpose of a Matrix

The transpose of a matrix is obtained by swapping its rows with its columns. In other words, the rows of the original matrix become the columns of the new matrix, and the columns become the rows.

A matrix is a rectangular arrangement of numbers or functions, organized into rows and columns. These numbers or functions are called the elements or entries of the matrix. When transposing a matrix, its dimensions change: if the original matrix has an order of m × n (rows by columns), the transposed matrix will have an order of n × m.

For example, consider a matrix A. To find its transpose A^T:

• The elements in the first row of A become the elements in the first column of A^T.
• The elements in the second row of A become the elements in the second column of A^T.

This operation reorganizes the structure of the matrix while preserving the data. The transposed matrix is denoted as A^T.

5.0Applications of Matrix Operations

1. Computer Graphics: Matrix transformations like scaling and rotation are used to manipulate images and models.
2. Physics: Representing systems of linear equations in mechanics and circuits.
3. Data Science: Matrices handle large datasets efficiently, enabling operations like regression analysis and eigenvector computation.

6.0Solved Problems

Problem 1: Find the value of x and y if

2  [ 1  3 ]     [ y  0 ]     [5  6  ] 

 [0   x]  + [1   2 ] =  [ 7  8 ]

Solution:

After equating the relevant elements of L.H.S. and R.H.S., we can quickly obtain the necessary values of x and y by applying the process of matrix multiplication and addition. 

We have,

2  [ 1  3 ]     [ y  0 ]      [5  6  ] 

[0   x]  + [1   2 ] =  [ 1  8 ]

⇒ [ 2  6 ]     [ y  0 ]      [5  6  ] 

 [0  2x]  + [1   2 ] =  [ 1  8 ]

⇒ [ 2 + y  6 + 0]     

[0 + 1  2x + 2]  = [5  6 ] 

[1   8]

Equating the corresponding elements, a₁₁ and a₂₂, we get

2 + y = 5 ⇒ y = 3

2x + 2 = 8  ⇒ 2x = 6 ⇒ x = 3

Hence, x = 3 and y = 3.

Problem 2: Matrix Multiplication

Given A = [ 1  2 ]

[ 3  4 ]

And B =  [ 5  6 ] 

[ 7  8 ]

Find AB.

Solution:

AB = [(1x5 + 2x7) (1x6 + 2x8)] 

[(3x5 + 4x7) (3x6 + 4x8)]

= [19 22] 

[43 50]

These matrices and matrix operations are the cornerstones of linear algebra, forming an integral tool for a number of mathematical and practical applications. To multiply the operations of matrices, to carry out elementary matrix operations, and to use such other related tools as the transpose of a matrix and its conjugate transpose forms powerful problem-solving tools and means of modeling the real world.