Determinant

Solution:

The determinant of order n is a number D, created from n² numbers a_ik, arranged into a square table with n rows and n columns of the form

a₁₁,a₁₂,...a_1n => row of the determinant

a₁₁,a₂₁,...a_n1 => column of the determinant

a₁₁,a₂₂,...a_nn => main diagonal

Value of the determinant:

• Second-order determinant

• Third-order determinant (Sarrus' rule)

Determinant of degree n:

A determinant of degree n has n rows and n columns (n > 3). The subdeterminant (minor) M_ij corresponding to the element a_ij is obtained from determinant D by removing the i-th row and j-th column.

Algebraic complement (cofactor):

A_ij = (-1)^i+j · a_ij · M_ij

Expansion of a determinant:

a.) along the i-th row:

D = a_i1M_i1 + a_i2M_i2 + ... + a_inM_in

b.) along the j-th column:

D = a_1jM_1j + a_2jM_2j + ... + a_njM_nj

Solution:

• 1)    The value of the determinant does not change if we swap its rows with its columns (i.e., transpose).
• 2)    The value of the determinant does not change if we add (or subtract) any multiple of another row to a given row.
• 3)    The value of the determinant equals zero if all elements in some row are zero.
• 4)    The value of the determinant equals zero if some row is a scalar multiple of another row.
• 5)    The value of the determinant equals zero if two rows are identical.
• 6)    The determinant changes sign if we interchange two rows of the determinant.
• 7)    We can factor out a number “c” from the determinant that multiplies some row.
• 8)    Multiplying the determinant by a number “c” means multiplying one of its rows by that number (only a single row). c <> 0
  

Note:

The stated properties also apply to the columns of the determinant.

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The area of triangle ABC is 7 j²

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The area of triangle ABC is 19 j²

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The condition for points A[x_a;y_a], B[x_b;y_b], C[x_c;y_c] to lie on the same line is

D = 0 => Points A, B, C lie on the same line

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D <> 0 => The points do not lie on the same line

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Let points A[x_a;y_a], B[x_b;y_b], X[x,y] lie on line p, then

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Cramer's rule 1.

The system has no solution

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Cramer's rule 2.

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Expansion along the fourth row:

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Condition:

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Expansion along the third row:

Points A, B, C, D lie in the same plane.

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Equation of the plane:

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Expansion along the second row:

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Expansion along the fourth row:

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