Properties

• i) det(A) = det(A^t). Then, all properties applying to rows are true for columns, too.

• ii) If we exchange two rows or two columns, the sign determinant changes:

Consequently, if we make an odd number of changes the sign of the determinant will change but it won’t if the number of changes is even.

Example:

Example:

• iv)

Analogous for columns

Demonstration: if we develop the determinant for this column:

|A|= A_1j(b₁ + c₁) +…= A_1j·b₁+…+A_1jc₁ QED

• v) If A has a column or a row of zeros, the determinant is 0

• vi) If A has two equal rows or columns → |A| = 0

Demonstration: we can suppose that R_i = R_j

• vii) If two rows or columns are proportional → |A| = 0

• viii) If we add a row by a number, k·R_i , to another row, R_j, the determinant doesn’t change.

Demonstration: |A| = |…F_i…F_j…|

|...F_i…F_j+kF_i…| = |..F_i…F_j…|+ |..F_i…kF_i…| = |A|

• ix) |A·B| = |A|·|B|

NOTES:

1) If all the elements in a row or column are zero except for one, the determinant is the product of that element by its cofactor.

2) If the matrix is triangular or diagonal, the determinant is the product of the elements in its diagonal.

3) If a row or column is a linear combination of the others   (R_i = k₁R₁+k₂R₂+…)→ |A| = 0

Examples:

1) If

calculate:

2) Solve

3) Solve

Exercises:

1.- Calculate:

2.- Solve:

3.- If

calculate:

Solutions: 1) a) -36; b) -56; 2) a) x € {-1,2,3}; b) a=b, b=c, a=c; 3) 36