Determinants

Determinants of a Matrix:

  1. Definition:

    • For a square matrix A, the determinant det(A) is a scalar value calculated from its elements.
    • It is denoted by vertical bars or det(A) or A.
  2. Calculation:

    • For a 2×2 matrix A=[abcd], det(A)=adbc.
    • For larger matrices, the calculation involves expansion by minors or other methods like row reduction.

For a 3x3 matrix A of the form:

A=[abcdefghi]

The determinant det(A) is computed using the following formula:

det(A)=aei+bfg+cdhcegbdiafh
 
Expansion by Minors:

Another way to calculate the determinant of a 3×3 matrix is by using expansion by minors. It involves breaking down the matrix into smaller determinants based on cofactors.

For matrix A:

A=[abcdefghi]

The determinant det(A) can be calculated as:

det(A)=adet([efhi])bdet([dfgi])+cdet([degh]) Where det([xyzw])=xwyz.

Properties of Determinants:

  1. Multiplicative Property: det(AB)=det(A)×det(B) for square matrices A and B.

  2. Transpose Property: det(A)=det(AT) for any square matrix A.

  3. Inverse Property: If Ais invertible, then det(A1)=1det(A).

  4. Scalar Multiplication: det(kA)=kn×det(A) for a square matrix A of order n and scalar k.

Significance and Applications:

  1. Matrix Inverses: A square matrix A is invertible if and only if det(A)0.

  2. System Solvability: For a system of linear equations AX=B, the system has a unique solution if and only if det(A)0.

  3. Geometric Interpretation: The absolute value of the determinant represents the scaling factor in transformations and encodes information about the orientation change and volume scaling in space.

Cramer's Rule:

Cramer's rule is a technique to solve systems of linear equations using determinants. For a system AX=B where A is a square matrix and det(A)0:

  • The solution for xi can be expressed as xi=det(Ai)det(A) where Ai is obtained by replacing the ith column of A with B.

Limitations:

  • Calculating determinants becomes computationally expensive for larger matrices due to expansion by minors or other methods.