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THE DETERMINANT OF A MATRIX

Definition

Definition:

2 x 2 Matrices

3 x 3 Matrices

4 x 4 Matrices

Example (2 x 2 Matrix)

A = [4, 2 ; -3 , 1]

det(A) = (4)*(1) - (2)*(-3)

      = 4-(-6)
      = 10

Example (3 x 3 Matrix)

A = [2, 5, 1 ; 3, 2, 2 ; 1, 5, 2]

det(A) = 2*[(2*2) - (2*5)] - 5*[(3*2) - (2*1)] + 1*[(3*5) - (2*1)]

      = 2*(4 - 10) - 5*(6 - 2) + 1*(15 - 2)
      = 2*(-6) - 5*(4) + 1*(13)
      = -12 - 20 + 13
      = -19

Example (4 x 4 Matrix)

A = [2, 3, 7, 2 ; 1, 6, 2, 3 ; 1, 1, 4, 3 ; 4, 6, 5, 1]

det(A) = 2*{6*[(4*1) - (3*5)] - 2*[(1*1) - (3*6)] + 3*[(1*5) - (4*6)]} - 3*{1*[(4*1) - (3*5)] - 2*[(1*1) - (3*4)] + 3*[(1*5) - (4*4)]} + 7*{1*[(1*1) - (3*6)] - 6*[(1*1) - (3*4)] + 3*[(1*6) - (1*4)]} - 2*{1*[(1*5) - (4*6)] - 6*[(1*5) - (4*4)] + 2*[(1*6) - (1*4)]}

      = 2*[6*(4-15) - 2*(1-18) + 3*(5-24)] - 3*[1*(4-15) - 2*(1-12) + 3*(5-16)] + 7*[1*(1-18) - 6*(1-12) + 3*(6-4)] 
        - 2*[1*(5-24) - 6*(5-16) + 2*(6-4)]
      = 2*[(6*(-11)) - (2*(-17)) + (3*(-19))] - 3*[(1*(-11)) - (2*(-11)) + (3*(-11))] + 7*[(1*(-17)) - (6*(-11)) + (3*2)] - 2*[(1*(-19)) - (6*(-11)) + (2*2)]
      = 2*(-66 - (-34) + (-57)) - 3*(-11 - (-22) + (-33)) + 7*((-17) - (-66) + 6) - 2*((-19) - (-66) + 4)
      = 2*(-89) - 3*(-22) + 7*(55) - 2*(51)
      = -178 - (-66) + 385 - 102
      = 171

Important Properties of the Determinant

- The determinant of the transpose of a matrix will be the same as the determinant of the original matrix.

- If two rows in the matrix are interchanged, the determinant will be the negative of the determinant of the original matrix.

- If two rows or columns in a matrix are the same, the determinant will be zero.

- If there is a row of zeros in the matrix, the determinant will be zero.

- If a row or column in a matrix was multiplied by a real number, then the determinant will be that real number multiplied by the determinant of the original matrix.

- If the matrix is either an upper or a lower triangular matrix, the determinant will be the product of the elements on the main diagonal.

- If two matrices are n x n, then the determinant of one multiplied by the determinant of the other is the determinant of the full matrix of the two next to each other.


Nonsingular and Similar Matrices

Definition: A square matrix is called nonsingular if its determinant is not equal to zero.

- For square matrices, then the linear combination of the matrix has a nontrivial solution if and only if the matrix is NOT nonsingular.

- For nonsingular matrices, the determinant of the inverse of that matrix is equal to 1 over the determinant of the matrix.

- If two matrices are similar matrices, then the determinants of the two are equal.



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