Determinants-Expand Along Matrices

A matrix is a rectangular table with rows and columns of numbers, symbols, or expressions. It constitutes a mathematical object or a property of such an object. The horizontal arrays and vertical collections are called rows and columns, respectively. The numbers in the rows and columns are known as elements or entities.

An example of a matrix is

This is a matrix with two rows and three columns, also referred to as a 2X2 matrix.

To calculate determinants, one has to expand along with matrices. The method of evaluating a determinant by expanding along a row or a column (matrices) is called Laplace Expansion or Cofactor Expansion.

Classification of matrices

Based on order and elements, matrices are classified as:

1. Zero matrix
2. Column matrix
3. Row matrix
4. Square matrix
5. Identity matrix
6. Scalar matrix
7. Diagonal matrix

• Zero matrix

It is also called a null matrix, as all the entries in this matrix are zero.

Example:

• Column matrix

It is also called a column vector, as it comprises a single column with m elements.

Example:

• Row matrix

It is also called row vector, as it comprises a single row with m elements.

Example:

• Square matrix

This matrix has an equal number of rows and columns.

Example: square matrix of the order 2.

Square matrix of the order 3. 

•  Identity matrix

It is a square matrix that has the principal diagonal elements as 1 and others as 0.

Example:

Identity matrix of the order 2

Identity matrix of order 3

•  Scalar matrix

It is a square matrix that has all its diagonal elements equal and all the off-diagonal elements zero. 

Example:

• Diagonal matrix 

A square matrix with a diagonal part with non-zero elements running from the upper left to the lower right.

Example:

What is a determinant? 

For a square matrix, a determinant can be defined in different ways.

• To formulate a determinant, expand along matrices taking into account the top row elements and the corresponding minors.
• The first element of the top row is multiplied by its minor.
• The product of the second element is subtracted from its minor.
• Alternate addition and subtraction of the product of each element. Begin from the top row with respective minor till every element of the top row is included.

For example:

Find out the determinant of the following

Solution:

• Cross multiply the rows with columns 
• Multiplication results in the following

 5 x 1 = 5

7 x 3= 21

• Subtraction of the products

 5- 21

 -16

The value of the determinant is – 16.

2 X 2 matrix determinant 

A 2×2 matrix comprises two rows and two columns.

This is an example of a 2×2 matrix determinant.

Expand along matrices to determine the determinant of the above matrix.

• Multiplying the cross rows and column.

 a x d= ad

 b x c= bc

• Subtraction of the products.

Ad – bc

• The value of the determinant is ad- bc.

3 X 3 matrix determinant 

A 3 X 3 matrix has three rows and three columns.

This is an example of a 3 X 3 matrix determinant.

Expand along the matrices to determine the determinant of the above matrix.

•  First, expand any one row, and the determinant solution can be derived.
• The expanded version will be 

• The 2 X 2 determinants will be solved as mentioned in the 2 X 2 matrix determinant.
• After solving that, the simple multiplication of row 1 can lead to the next step.
• When determining a determinant in a 3×3 matrix, the signs are  +, -, +  ( alternate).
• Following all these steps will give the final determinant.

Operation on matrices

The three basic operations on matrices are addition, scalar multiplication, and transposition. That means there are three ways to do it on the rows of a matrix. 

Addition 

Sum A+B of two m-by-n matrices A and B is calculated as 

(A + B)i,j = Ai,j + Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.

Scalar multiplication

The product of a number represented as cA, c (also called scalar) and a matrix A is obtained by multiplying every entry of A with c:

(cA)i,j = c · Ai,j.

Transposition

An m-by-n matrix A transpose is n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns. 

(AT)i,j = Aj,i.

Conclusion 

The term “matrix” was used by Bertrand Russell and Alfred Whitehead in their Principia Mathematica of the axiom of reducibility, which proposed this principle to reduce any function to one of a lower type.

Numbers to expand along matrices can represent data and can be used to represent mathematical equations. They are found to be beneficial in engineering as well. Solving issues by multiplying matrices give quick and good approximations of complicated calculations.