matrix operations: determinant
TRANSCRIPT
Matrix Operations: Determinant
Determinants
β’ Determinants are only applicable for square matrices.
β’ Determinant of the square matrix π΄ is denoted as:
det(π΄) or π΄
β’ Recall that the absolute value of the determinant of a 2 Γ 2 matrix is equal to the area of parallelogram of the rows of that matrix.
β’ Similarly, the absolute value of the determinant of a 3 Γ 3 matrix is equal to the volume of parallelepiped of the rows of that matrix.
β’ Therefore, the absolute value of the determinant of a π Γ π matrix is equal to the n-dimensional volume, constructed by the rows of that matrix.
Determinant of a 2 Γ 2 matrix
β’ Recall that:
π΄ =π11 π12π21 π22
, π΄ =π11 π12π21 π22
= π11π22 β π12π21.
π1
π2
ππππ
Determinant of a 3 Γ 3 matrix
β’ Also recall the determinant for a 3 Γ3 matrix:
β’ π =
π11 π12 π13π21 π22 π23π31 π32 π33
β’ If the row vectors are linearlydependent, then the determinantis zero, and the matrix is NOT invertible.β’ Notice if the row vectors arelinearly dependent the volumewill be zero, as the vectors lie on a plane on a line.
π1π2π3
Determinant of a 3 Γ 3 matrixβ’ To compute the determinant of a 3 Γ 3 matrix,.
β’ The first element in the top row is multiplied with the determinant of the sub-matrix resulting from removing the (first) row and the (first) column corresponding to that element from the matrix.
β’ The negate of second element in the top row is multiplied with the determinant of the sub-matrix resulting from removing the (first) row and the (second) column corresponding to that element from the matrix.
β’ The third element in the top row is multiplied with the determinant of the sub-matrix resulting from removing the (first) row and the (third) column corresponding to that element from the matrix.
β’ π =
π11 π12 π13π21 π22 π23π31 π32 π33
= π11π22 π23π32 π33
β π12π21 π23π31 π33
+ π13π21 π22π31 π32
=
π11 π22π33 β π23π32 β π12 π21π33 β π23π31 + π13 π21π32 β π22π31
Determinant of a 3 Γ 3 matrix / Cofactor
β’ In the determinant of a 3 Γ 3 matrix, we multiplied the first row elements in their corresponding cofactors.
β’ The cofactor of the element π, π of π Γ π matrix π΄ is:πΆππ = (β1)π+πdetπππ
β’ Where πππ is submatrix after removing row π and column π.β’ Determinant of π΄ is:
detπ΄ = ππ1πΆπ1 + ππ2πΆπ2 +β―+ ππππΆππβ’ In the above formula the row π could be any row of π΄ and it is not
necessarily the first row.β’ In fact it need not be a row. It can be any column j. β’ (So in order to compute the determinant, it is always wise to choose the
row or a column that has most number of zeroes and compute the cofactor of only its non-zero elements.)
Determinant properties
β’ The determinant of identity matrix is 1.πΌ = 1
β’ The determinant changes sign when two rows are exchanged.π ππ π
= βπ ππ π
β’ The determinant is a linear function of each row separately.π‘π π‘ππ π
= π‘π ππ π
π + πβ² π + πβ²
π π=
π ππ π
+πβ² πβ²
π π
Determinant properties
β’ If one row is a scalar multiple of another row then det(π΄) = 0
π ππ‘π π‘π
= 0π π ππ π ππ‘π π‘π π‘π
= 0
π π ππ π π
π + π π + π π + π= 0,
π π ππ π π
2π + π 2π + π 2π + π= 0
π π ππ π π
2π + 5π 2π + 5π 2π + 5π= 0
Determinant properties
β’ Row reduction does not change the determinant of π΄π π
π β πΎπ π β πΎπ=
π ππ π
πΎ is a non-zero scalar
β’ A matrix with a row of zeros has det(π΄) = 0π π0 0
= 0
Determinant properties
β’ If π΄ is a triangular then the determinant is the product of diagonal elements.
π π0 π
= ππ,π 0π π
= ππ
This is also applicable for diagonal matrices:π 0 00 π 00 0 π
= πππ
β’ If π΄ is singular (columns or rows are linearly dependent) det(π΄) = 0
β’ π΄π΅ = π΄ π΅
β’ π΄π = π΄
Rank of Matrix
β’ Let π = min πππ€, ππππ’ππ
β’ Rank of matrix is the size of the largest square sub-matrix with non-zero determinant.
β’ Matrix is full-ranked, if its rank = m.
β’ Matrix is rank-deficient, if its rank < m.
β’ It is not possible to have matrixβs rank > m.
Sub-Matrix
β’ In order to find the rank of matrix we should find the largest quaresub-matrix with non-zero determinant.
β’ For making a sub-matrix we are allowed to remove rows or columns of a matrix
β’ Example: A is a 5 Γ 3 matrix
β’ Removing two rows of Aπππ€1πππ€2πππ€3πππ€4πππ€5
=πππ€2πππ€4πππ€5
Matrix Rank
β’ Example: Find the rank of matrix A
π΄ =0 1 21 2 12 7 8
Row 1 and Row 2 of matrix A are linearly independent. However Row 3 is a linear combination of Row 1 and 2.
πππ€3 = 3 Γ πππ€1 + 2 Γ πππ€2
So A only have two independent row vectors. Now let remove third row and first column of A then we have a 2 Γ 2 matrix which determinant is not zero.
1 22 1
β 0
So rank of A is 2.