What are the properties of determinants

Determinant

Properties of a determinant

Feature 1
The determinant of a matrix and the determinant of its transpose are identical

\ (| A | = | A ^ T | \)

Property 2
If you swap two rows (or two columns) of a matrix, the sign of the determinant changes. If you swap three rows (or three columns), the sign does not change.

\ (\ begin {vmatrix} {\ color {blue} a_1} & {\ color {blue} a_2} & {\ color {blue} a_3} \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \ end { vmatrix}
= - \ begin {vmatrix} b_1 & b_2 & b_3 \ {\ color {blue} a_1} & {\ color {blue} a_2} & {\ color {blue} a_3} \ c_1 & c_2 & c_3 \ end { vmatrix}
= - \ left [- \ begin {vmatrix} b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \ {\ color {blue} a_1} & {\ color {blue} a_2} & {\ color {blue} a_3 } \ end {vmatrix} \ right] \)

Feature 3
If you multiply a row (or a column) of the matrix by a number, the determinant is also multiplied by this number.

\ (det (D) = \ begin {vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \ end {vmatrix} \)

\ (\ begin {vmatrix} {\ color {blue} \ lambda} \ cdot a_1 & {\ color {blue} \ lambda} \ cdot a_2 & {\ color {blue} \ lambda} \ cdot a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \ end {vmatrix}
= \ begin {vmatrix} {\ color {blue} \ lambda} \ cdot a_1 & a_2 & a_3 \ {\ color {blue} \ lambda} \ cdot b_1 & b_2 & b_3 \ {\ color {blue} \ lambda } \ cdot c_1 & c_2 & c_3 \ end {vmatrix} = {\ color {blue} \ lambda} \ cdot det (D) \)

Feature 4
The determinant of a product of two matrices corresponds to the product of their determinants.

\ (| A \ cdot B | = | A | \ cdot | B | \)

Feature 5
A determinant is zero if

  • a row / column consists only of zeros
  • two rows / columns are the same
  • a row / column is a linear combination of other rows / columns

Applications