Matrix Operations: Determinant

Transcription

1 Matrix Operations: Determinant

2 Determinants Determinants are only applicable for square matrices. Determinant of the square matrix A is denoted as: det(a) or A Recall that the absolute value of the determinant of a 2 2 matrix is equal to the area of parallelogram of the rows of that matrix. Similarly, the absolute value of the determinant of a 3 3 matrix is equal to the volume of parallelepiped of the rows of that matrix. Therefore, the absolute value of the determinant of a n n matrix is equal to the n-dimensional volume, constructed by the rows of that matrix.

3 Determinant of a 2 2 matrix Recall that: a a 11 a 12 A = a 21 a, A = a 1 11 a 12 a 2 22 a 21 a = a 11 a 22 a 12 a a 2 a 1

4 Determinant of a 3 3 matrix Also recall the determinant for a 3 3 matrix: r 1 r 11 r 12 r 13 R = r 2 r 21 r 22 r 23 r 3 r 31 r 32 r 33 If the row vectors are linearly dependent, then the determinant is zero, and the matrix is NOT invertible. Notice if the row vectors are linearly dependent the volume will be zero, as the vectors lie on a plane on a line.

5 Determinant of a 3 3 matrix To compute the determinant of a 3 3 matrix,. 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. 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. The third element in the top row is multiplied with the determinant of the submatrix resulting from removing the (first) row and the (third) column corresponding R = to that element from the matrix. r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 = r 11 r 22 r 23 r 32 r 33 r 12 r 21 r 23 r 31 r 33 + r 13 r 21 r 22 r 31 r 32 = r 11 r 22 r 33 r 23 r 32 r 12 r 21 r 33 r 23 r 31 + r 13 r 21 r 32 r 22 r 31

6 Determinant of a 3 3 matrix / Cofactor In the determinant of a 3 3 matrix, we multiplied the first row elements in their corresponding cofactors. The cofactor of the element i, j of n n matrix A is: C ij = ( 1) i+j det M ij Where Mij is submatrix after removing row i and column j. Determinant of A is: det A = a i1 C i1 + a i2 C i2 + + a in C in In the above formula the row i could be any row of A and it is not necessarily the first row. In fact it need not be a row. It can be any column j. (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.)

7 Determinant properties The determinant of identity matrix is 1. I = 1 The determinant changes sign when two rows are exchanged. c d a b = a b c d The determinant is a linear function of each row separately. ta tb a b = t c d c d a + a b + b c d = a b c d + a b c d

8 Determinant properties If one row is a scalar multiple of another row then det(a) = 0 a b c a b ta tb = 0 d e f = 0 ta tb tc a b c d e f a + d b + e c + f = 0, a b c d e f 2a + 5d 2b + 5e 2c + 5f a b c d e f 2a + d 2b + e 2c + f = 0 = 0

9 Determinant properties Row reduction does not change the determinant of A a b a b = c γa d γb c d γ is a non-zero scalar A matrix with a row of zeros has det(a) = 0 a b 0 0 = 0

10 Determinant properties If A is a triangular then the determinant is the product of diagonal elements. a b 0 d = ad, a c 0 d = ad This is also applicable for diagonal matrices: a b 0 = abc 0 0 c If A is singular (columns or rows are linearly dependent) det(a) = 0 AB = A B A T = A

11 Rank of Matrix Let m = min row, column Rank of matrix is the size of the largest square sub-matrix with nonzero determinant. Matrix is full-ranked, if its rank = m. Matrix is rank-deficient, if its rank < m. It is not possible to have matrix s rank > m.

12 Sub-Matrix In order to find the rank of matrix we should find the largest quare sub-matrix with non-zero determinant. For making a sub-matrix we are allowed to remove rows or columns of a matrix Example: A is a 5 3 matrix Removing two rows of A row1 row2 row3 row4 row5 = row2 row4 row5

13 Matrix Rank Example: Find the rank of matrix A A = Row 1 and Row 2 of matrix A are linearly independent. However Row 3 is a linear combination of Row 1 and 2. row3 = 3 row1 + 2 row2 So A only have two independent row vectors. Now let remove third row and first column of A then we have a 2 2 matrix which determinant is not zero So rank of A is 2.