Maths Determinants: Overview, Questions, Preparation

Determinant is a scalar value that can be calculated from a square matrix (a table with equal number of rows and columns). It is a technique of associating a single unique numerical value to a square matrix. It can help in providing useful information related to the matrix such as whether the matrix is singular or invertible.

Mathematical Representation:

If A = [ a b c d ] 

Determinant of matrix A:

$\left|A\right|=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|$= det.(A) and it’s value will be calculated by “ad – bc”

Where,

$a_{ij}=(i,j){\text{th}}^{}$element of A

Some crucial concepts of this chapter Determinants are given below for reference which the candidates need to be familiar with:

1. Minors

Minors of an element is referred to the determinant obtained by deleting its i^th row and j^th column in which lies.

$\text{Minor of}{\mathrm{ a}}_{ij}\text{ can be denoted by}{\mathrm{ M}}_{ij}$

Where,

I = row

J = column

2. Cofactors

A cofactor is a concept similar to that of minors, but with a simple difference of a sign. The cofactor of an element can be denoted by the equation:

$C_{ij}={(-1)}^{i+j}{M}_{ij}$

$\text{Where, }{M}_{ij}=\text{minor of}{\mathrm{ a}}_{ij}$

If you want to calculate the cofactor, simply calculate the minor and add the sign factor to compute the result, which will be the cofactor.

 

3. Area of a Triangle

Area of a triangle ABC whose vertices are $A (x_1,y_1),\mathrm{ B} (x_2,y_2)\text{, and }C (x_3,y_3)$can be easily calculated using the formula:

$\Delta =\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$

• If the area of $△\text{ABC}=0$ , the points are known to be collinear.

4. Singular matrix

A singular matrix is referred to the square matrix whose determinant is zero. It can be denoted as $\left|\mathit{A}\right|=0$

 

5. Non-singular matrix

A non-singular matrix is referred to the square matrix whose determinant is not equal to zero i.e.    0. A square matrix A will be invertible if and only if A is a non-singular matrix.

A determinant can be expanded along any row/column and the result in all cases will be the same. The procedure to do so is known as Laplace expansion that simply means expressing the value as a sum of the products of the elements and their cofactors.

Fun Fact: Applying the formula of Laplace expansion over a row/column with more number of zeros will ease the calculation process with fewer chances of errors.

Determinants can be categorized into various types according to their properties and arrangement of elements. These are mentioned below.

1) By Order/Size:

1. First-order determinant: This is a matrix of 1x1 order. The element only is the determinant value of the matrix. 
2. Second-order determinant: This is a matrix of 2x2 order. The determinant value will be ad - bc [[a, b], [c, d]],. 
3. Third-order determinant: This is a matrix of 3x3 order. The determinant value is calculated by expanding along a row or column. 
4. N-order determinant: This is a matrix of nxn order. The determinant value is calculated through same way as third order. 

 2) Special Determinants:

1. Symmetric Determinant: A determinant is symmetric if its corresponding matrix is also the same when transposed. In a symmetric matrix, elements aᵢⱼ and aⱼᵢ are equal for all i and j i.e, aij=aji​. 
2. Skew-symmetric Determinant: A determinant is skew-symmetric if its corresponding matrix is also skew-symmetric. Major difference between symmetric and skew-symmetric matrices is that in the latter one, aᵢⱼ = -aⱼᵢ for all i and j, and all diagonal elements (aii) are zero. 
3. Diagonal Determinant: Here all the non-diagonal elements of a matrix are 0 and the determinant value is the product of the diagonal elements.

$$Adjoint of a matrix is the transpose of the cofactor matrix of a given square matrix.

Let A = a i j be a square matrix.

Assume B = A i j

Where, A i j = cofactor of the elements a i j in matrix A.

The transpose B T of matrix B will be called as the adjoint of matrix A.

It can be denoted by "adj. A".

To find the adjoint of 2 x 2 matrix, simply use this trick:

A =                         a           b                           c           d                       :. adj. A =                         d           - b                           - c           a                    

 

Inverse of a matrix is simply the reciprocal of the original square matrix.

For a square matrix A, its inverse is a matrix A - 1 such that:

A ⋅ A - 1 = A - 1 ⋅ A = I

Algorithm to find A - 1 by Determinant method:

STEP 1: Find A .

STEP 2: If A = 0, then write "A is a singular matrix and hence cannot be invertible".

STEP 3: Calculate the cofactors of elements of matrix A.

STEP 4: Write the matrix of cofactors of elements of A and then obtain its transpose to get adj. A (i.e., adjoint A).

STEP 5: Find the inverse of A by using the relation A - 1 = 1 A adj. A .

 

Here are a few important applications of determinants in relation to mathematics which ideally you should be aware of.

1. Solving Linear Equations
2. Matrix Invertibility 
3. Area and Volume Calculations
4. Computer Graphics
5. Physics
6. Engineering