Matrices

Matrices are like a math table with numbers arranged in rows and columns. They are boxes filled with numbers, variables, and symbols. Each number inside a box is called an element of the matrix. The size of the matrix is called its order, that is, the number of rows and columns it has.

For example, if the matrix has 3 rows and 4 columns, we say its order is 3 × 4. 
 

Based on the order of the matrix, elements, and with certain conditions like equal number of rows and columns or all elements being zero, etc., matrices are represented in different types. Let’s discuss some of the basic types of matrices below.

• Singleton Matrix
• Row Matrix
• Column Matrix
• Square Matrix
• Zero Matrix
• Diagonal Matrix
• Identity Matrix
• Rectangular Matrix

Singleton Matrix: A singleton matrix consists of only one element. In this, the number of rows and columns is equal to 1. An example of a singleton matrix is [5]; here, the matrix has only one element. 

Row Matrix: The matrix that consists of only one row is called a row matrix.  The row matrix can have n number of elements, but it should be arranged in only one row. [3 5 7] is an example of a row matrix.

Column Matrix: A matrix with only one column is called a column matrix. Like a row matrix, a column matrix can also have n elements, but it should be arranged in a column. The example of a column matrix is: 

In the above example, the elements are arranged only in columns, so it is called a column matrix.

Square Matrix: A matrix with the same number of rows and columns is known as a square matrix. 

In the above matrix, the number of rows and the number of columns are equal. Therefore, it is a square matrix.

Zero Matrix: The zero matrix is also known as a null matrix. As the name implies, all the elements in the zero matrix are 0. 

Diagonal Matrix: A square matrix, where all the elements are zero except the diagonal elements, is called a diagonal matrix. 

In the above matrix, only the diagonal elements are non-zero, and all the other elements are zero. So, it is a diagonal matrix.

Identity Matrix: Identity matrix is a special type of diagonal matrix where all the diagonal elements are 1 and everything else is 0. 

Here, only the diagonal elements are 1, and all the other elements are 0, so it is an identity matrix.

Rectangular Matrix: A matrix that has a different number of rows and columns is called a rectangular matrix. 

In the above matrix, rows and columns are not equal. So, it is a rectangular matrix.
 

The calculations that are performed on matrices for solving problems are known as matrix operations. The main operations on matrices are,

• Matrix Addition
• Matrix Subtraction
• Scalar Multiplication
• Matrix Multiplication
• Transpose of Matrix
• Inverse of Matrix

Matrix Addition: The process of adding one matrix to another is known as matrix addition. We can only add the same-sized matrices by adding their matching elements. Add the first elements from both the matrices, then the second, and so on. 
Example:

Matrix Subtraction: Like addition, subtraction is also only done when the matrices are of the same size. We can subtract the first elements from both matrices, and then the second, and so on. Let’s see the subtraction using the below example.

Scalar Multiplication: Multiplying each number in the matrix by the same given number is known as scalar multiplication. An example of scalar multiplication is,

Matrix Multiplication: In matrix multiplication, we have to multiply the numbers in the first column of the first matrix with the numbers in the first row in the second matrix. 
Example: 

Transpose of a Matrix: Transposing a matrix means swapping its rows and columns. In the transposed matrix, each row becomes a column, and each column becomes a row. 
Example: 

Inverse of a Matrix: Inverse of a matrix can be done only when the determinant of the matrix is not zero. The inverse of a matrix can be found using a formula:
 

A determinant is a special number that can be found only in a square matrix. It is used to solve equations or find the inverse of a matrix. If the given matrix is A, we can write the determinant as |A|. For finding the determinant, we use the formula as,

If the given matrix is in the form of A =  cadb

Then the determinant |A| = ad - bc.

Example: 
 

For finding the determinant for a 3 × 3 matrix, we use the following steps

1. Use only the top row.
2. For each number in the top row, cover its row and column, you will get a small 2 × 2 matrix. Find the determinant for the new 2 × 2 matrix.
3. Multiply each number in the top row with the determinant of the 2 × 2 matrix. 
4. Use the signs +, -, + and add all the three answers. 
Let’s look at an example of a 3 × 3 matrix. 
 

Cover the row and column of the first number 1, we will get a new 2 × 2 matrix. 
 

Now add the first number 1 with the new determinants value, 1 × -3 = -3
Do these steps for the other two numbers as well. 

Add: -2 × -6 = 12
 

Add: 3 × -3 = -9
Now add all the three numbers using the signs +, -, + in order: 
-3 + 12 - 9 = 0
Therefore, the value of the determinant is 0.
 

• We can use determinants only if the matrix is a square matrix. In other words, the matrices which have the same number of rows and columns.

• If we are flipping any two rows or any two columns, the sign of the determinant changes from positive to negative and vice versa.

• If any two rows or columns are the same, then the determinant of that matrix is 0.

• The determinant of the identity matrix is 1.

• If the whole row or column is all zeros, the determinant is 0.

• If we multiply a row by any number, the determinant gets multiplied by that number too.

• If we multiply two square matrices, the determinant of the result is the product of the two individual determinants.

A square matrix that has an inverse is known as an invertible matrix. This means, multiplying a matrix by its inverse results in the identity matrix. Suppose A is a square matrix and B is the same size as the size of the A matrix, such that:

AB = BA = I

Here, A is the matrix,

B is the inverse of A,

I is the identity matrix.

Invertible matrices work only in the square matrices, such as 2 × 2, 3 × 3 matrices.

Example: 

Now the inverse of the matrix A is,
 

While multiplying A and B, if we get an identity matrix, then the matrix A is an invertible matrix.

Here, we get an identity matrix, therefore A is an invertible matrix.

The inverse of a matrix is like the opposite of the matrix.  It is the same as inverting a number, the inverse of 5 is ⅕. Likewise, if A is the matrix, the inverse of A is A-1. The inverse of a matrix can be found using a formula:

We get the transpose of the matrix when we flip the rows and the columns. The first row becomes the first column, the second row becomes the second column and so on. The transpose of the matrix A is written as AT. If the given matrix is: 
 

Then the transpose A^T is:

The trace of the matrix is the sum of the numbers on the main diagonal of a square matrix. The sum of the numbers in the top left to the bottom right of the matrix. If A is the given matrix, we can find the trace of A using, Trace (A) = a11 + a22 + a33 + …

Example: 
If A =     ,

the trace is 3 + 2 = 5.
 

A minor of a matrix can be found when we remove one row and one column from a square matrix, and then find the determinant of the new matrix. Minor helps in calculating the determinant, cofactors, and inverse of a matrix. 
Example: For finding the minor in a 3 × 3 matrix, we should first find the determinant, and then the determinant of a smaller matrix formed by removing a specific row and column, that is called the minor. 

Here, if we find the determinant of the 1, we get a small 2 × 2 matrix,

(5 × 9) - (6 × 8) = 45 - 48 = -3.
So, the minor of 1 is -3.

The cofactor of an element is similar to its minor, but it also includes a sign based on the element's position in the matrix. It helps us find the determinant and the inverse of a matrix.

For finding the cofactor, we use the formula:

Cofactor = (-1)i + j × Minor

Here,’ i’ is the row index.

j is the column number.

Example: 
If we take a 3 × 3 matrix, 

For finding the cofactor 2, which is in row 1 and column 2. 
Find the determinant,

(4 × 9) - (6 × 7) = 36 - 42 = -6
So, the minor is -6.
Now add the sign using the formula,
Cofactor = (-1)^{i + j} × Minor
Cofactor = (-1)^{1 + 2} × (-6)
= (-1)³ × (-6)
= (-1) × (-6)
= 6.
Therefore, the cofactor of 2 is 6.
 

The adjoint of the square matrix is the transpose of the cofactor matrix. For finding the adjoint of the matrix, follow the steps given below,

1. Find the cofactor of each element in the matrix.
2. Make a new matrix using all the cofactors that are called cofactor matrices.
3. The transpose of the cofactor matrix is the adjoint of the matrix.

If A is a square matrix, then, Adj(A) = Transpose of the cofactor matrix of A. 

For solving a system of equation using matrixes, follow the steps given below,

Step 1: Make a Matrix
A matrix is just a table of numbers that represent your equations. Let’s see how to solve it using an example.
x + y = 5
x - y = 1

Using this equation, we have to create an augmented matrix. An augmented matrix is a matrix that includes the coefficients and constants from a system of linear equations. It helps to solve the system using methods like row operations.

Therefore, the matrix will be,

Step 2: Use Row Operations to Simplify
We want to make the bottom left number to 0. So, subtract row 1 from row 2. So the new matrix will become,

Step 3: Solve using Gaussian elimination method
Start with the second row:
-2y = -4
y = 4/2
y = 2

Now substitute the value of y to the first row.
x + y = 5
x + 2 = 5
x = 5 - 2
x = 3

Step 4: Check the Solution
Substitute the value of x and y to the given equations and check whether both the answers are matching.
x + y = 3 + 2 = 5
x - y = 3 - 2 = 1

Therefore, the answers are matching. So the values are: x = 3, y = 2

Matrices are widely used in different fields and professions. Here are some real life applications where matrices are used.

• Computer graphics and Animation: We use matrices to create and transform images in movies and games. They help move, rotate, and scale characters or objects by changing their position and orientation on the screen.

• Artificial Intelligence and Machine Learning: Matrices are used to process large amounts of data and train smart systems in machine learning and artificial intelligence. Face recognition in the phone uses matrices to process the pixel data for identifying the face.

• Medical Imaging: While processing the images from MRI or CT scans matrices are used. An MRI scan produces a grid of numbers in matrix form that represents an image of your brain.