Determinant of Matrix | BA Economics | Mathematics of Economics.

 The determinant of a matrix is a scalar value computed for a given square matrix. Let’s explore this mathematical concept:

1. Definition of Determinant of Matrix:

   • The determinant of a matrix is calculated using the elements of a square matrix.
   • It can be considered as the scaling factor for the transformation of a matrix.
   • Geometrically, the determinant represents the volume scaling factor of the linear transformation defined by the matrix.
   • It is also expressed as the volume of the n-dimensional parallelepiped crossed by the column or row vectors of the matrix.
   • The determinant is positive or negative based on whether the linear mapping preserves or changes the orientation of n-space.

2. Calculating the Determinant:

   • The determinant is defined only for square matrices (where the number of columns and rows are equal).
   • For a 2×2 matrix, the determinant can be represented as: [ \Delta = \begin{vmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{vmatrix} ] The value of the determinant is given by: [ \text{det}(A) = a_{11} \cdot a_{22} - a_{21} \cdot a_{12} ]
   • For larger matrices (e.g., 3×3, 4×4, or n×n), there are techniques to find the determinant.

3. Usefulness of Determinants:

• Determinants are useful in various mathematical applications:
• Solving systems of linear equations.
• Calculating the inverse of a matrix.
• Performing calculus operations.

For 2x2 Matrix 

If [A] = [a]  then its determinant is given as |a|  which is equal to the value enclosed in the matrix.

The value of thedeterminant of a 2 × 2 matrix can be given as

det A =

$\begin{array}{l}{�}_{11}\times {�}_{22}–{�}_{21}\times {�}_{21}\end{array}$

Let us take an example to understand this very clearly,

$\begin{array}{l}{�}_{11}\times {�}_{22}–{�}_{21}\times {�}_{\mathrm{21<span\; face="Poppins,\; sans-serif"\; style="font-size:\; 16px;\; text-align:\; left;">Let\; us\; take\; an\; example\; to\; understand\; this\; very\; clearly,</span>}}\end{array}$

Example 2: The matrix  is given by, A =

 

Solution: To find the determinant of [A], let us expand the determinant along row 1.

Therefore, det A =

⇒ |A| = 4 (0 – 15) + 3(2+3) + 5(5-0) ⇒ |A| = -20

Example 1

Find the determinant of the matrix $A=\left[\begin{array}{cc}4& 1\\ 0& 2\end{array}\right]$

example 3x3

 Thanks for your time