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Properties of Determinants of Matrices

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  • Difficulty Level : Hard
  • Last Updated : 30 May, 2019
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Determinant of a Matrix is a scalar property of that Matrix. Determinant is a special number that is defined for only square matrices (plural for matrix). Square matrix have same number of rows and columns.

Determinant is used to know whether the matrix can be inverted or not, it is useful in analysis and solution of simultaneous linear equations (Cramer’s rule), used in calculus, used to find area of triangles (if coordinates are given) and more. Determinant of a matrix A is denoted by |A| or det(A).

Properties of Determinants of Matrices:

  1. Determinant evaluated across any row or column is same.
  2. If all the elements of a row (or column) are zeros, then the value of the determinant is zero.
  3. Determinant of a Identity matrix (I_n) is 1.
  4. If rows and columns are interchanged then value of determinant remains same (value does not change). Therefore, det(A) = det(A^T), here A^T is transpose of matrix A.
  5. If any two row (or two column) of a determinant are interchanged the value of the determinant is multiplied by -1.
  6. If all elements of a row (or column) of a determinant are multiplied by some scalar number k, the value of the new determinant is k times of the given determinant. Therefore, If A be an n-rowed square matrix and K be any scalar. Then |KA| = K^n|A| .
  7. If two rows (or columns) of a determinant are identical the value of the determinant is zero.
  8. Let A and B be two matrix, then det(AB) = det(A)*det(B).
  9. If A be a matrix then, |A^n| = (|A|)^n.
  10. Determinant of Inverse of matrix can be defined as |A^-^1| = \dfrac{1}{|A|}.
  11. Determinant of diagonal matrix, triangular matrix (upper triangular or lower triangular matrix) is product of element of the principle diagonal.
  12. In a determinant each element in any row (or column) consists of the sum of two terms, then the determinant can be expressed as sum of two determinants of same order. For example,

  13. If B is obtained by adding c-times a row of A to a different row, the \det (B) = \det (A).
  14. Let A be a matrix then,
    |adj(A)| = (|A|)^n^-^1
    |adj(adj(A))| = |A|^(^n^-^1^)^*^(^n^-^1^),
    Here adj(A) is adjoint of matrix A.

  15. If value of determinant \Delta becomes zero by substituting x = \alpha, then x-\alpha is a factor of \Delta.
  16. Here, cij denotes the cofactor of elements of aij in \Delta.

  17. In a determinant the sum of the product of the elements of any row (or column) with the cofactors of the corresponding elements of any other row (or column) is zero. For example,
    d = ai1*Aj1 + ai2*Aj2 + ai3*Aj3 +…… + ain*Ajn, here Aj1, Aj2, Aj3 …Ajn are cofactors along elements of jth row.

  18. Let \lambda_1, \lambda_2, \lambda_3.......\lambda_n are the Eigenvalues of A (square matrix of order n). Then det(A) = \lambda_1*\lambda_2*\lambda_3.....*\lambda_n, product of Eigenvalues.

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