Summary

  • Determinant of a matrix A = [a₁₁] of order 1 is given by |A| = a₁₁.
  • Determinant of a matrix A = \(\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}\) of order 2 is given by \(|A| = a_{11}a_{22} - a_{12}a_{21}\).
  • Determinant of a matrix A = \(\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\) of order 3 is given by \(|A| = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2)\).
  • For any square matrix A, the area of a triangle with vertices \((x_1, y_1), (x_2, y_2)\) and \((x_3, y_3)\) is given by \(\frac{1}{2}|\begin{smallmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{smallmatrix}|\).
  • Minor of an element \(a_{ij}\) of the determinant of matrix A is the determinant obtained by deleting the \(i^{th}\) row and \(j^{th}\) column.
  • Cofactor of an element \(a_{ij}\) of the determinant of matrix A is \(A_{ij} = (-1)^{i+j} M_{ij}\).
  • Value of determinant of a matrix A is obtained by the sum of the product of elements of a row (or a column) with their corresponding cofactors.
  • If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero.
  • Adjoint of A, denoted by adj A, is the transpose of the cofactor matrix.
  • \(A(adj A) = (adj A)A = |A|I\).
  • A square matrix A is said to be singular or non-singular according as \(|A|=0\) or \(|A| \neq 0\).
  • If AB = BA = I, where B is the inverse of A, then \(A^{-1}=B\).
  • \(A^{-1} = \frac{1}{|A|}(adj A)\).
  • A system of linear equations is consistent if it has one or more solutions, and inconsistent if its solution does not exist.
  • For a system of equations \(AX=B\):
    • If \(|A| \neq 0\), there is a unique solution given by \(X=A^{-1}B\).
    • If \(|A| = 0\) and \((adj A)B \neq 0\), then there is no solution.
    • If \(|A| = 0\) and \((adj A)B = 0\), then the system has either no solution or infinitely many solutions.