Cs130 day03

Determinants

CS 130: Mathematical Methods in Computer Science Linear Systems and Matrices Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph Day 3

Determinants

Linear Equations and Matrices

Determinants Properties of Determinants Cofactor Expansion The Inverse of a Matrix Solving Linear Systems

Determinants

Linear Systems and Matrices

Determinants Properties of Determinants Cofactor Expansion The Inverse of a Matrix Solving Linear Systems

Determinants

Determinant Let A = [aij ] be an n × n matrix. We define the determinant of A (written det(A) or |A|) by X det(A) = |A| = (±)a1j1 a2j2 · · · anjn , where the summation ranges over all permutations j1 j2 · · · jn of the set S = {1, 2, · · · , n}. The sign is taken as + or − according to whether the permutation j1 j2 · · · jn is even or odd.

Note: Let S = {1, 2, · · · , n} be the set of integers from 1 to n, arranged in ascending order. A rearrangement j1 j2 · · · jn of the elements of S is called a permutation of S. A permutation j1 j2 · · · jn of S = {1, 2, · · · , n} is said to have an inversion if a larger integer jr precedes a smaller one js . A permutation is called even or odd according to whether the total number of inversions in it is even or odd.

Determinants

Examples

2 Ă— 2 matrix

A=

a11 a21

a12 a22

Determinants

Examples

2 Ă— 2 matrix

A=

a11 a21

a12 a22

det(A) = a11 a22 − a12 a21

Determinants

Examples

2 × 2 matrix

A=

a11 a21

a12 a22

det(A) = a11 a22 − a12 a21

3 × 3 matrix 

a11 B =  a21 a31

a12 a22 a32

 a13 a23  a33

Determinants

Examples

2 × 2 matrix

A=

a11 a21

a12 a22

det(A) = a11 a22 − a12 a21

3 × 3 matrix 

a11 B =  a21 a31

a12 a22 a32

 a13 a23  a33

det(B) = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 −a11 a23 a32 − a12 a21 a33 − a13 a22 a31

Determinants

Some Properties The determinants of a matrix and its transpose are equal, that is, det(AT ) = det(A) where the transpose of A is obtained by interchanging the rows and columns of A.

Determinants

Some Properties The determinants of a matrix and its transpose are equal, that is, det(AT ) = det(A) where the transpose of A is obtained by interchanging the rows and columns of A.

If matrix B results from matrix A by interchanging two rows(columns) of A, then det(B) = − det(A).

Determinants

Some Properties The determinants of a matrix and its transpose are equal, that is, det(AT ) = det(A) where the transpose of A is obtained by interchanging the rows and columns of A.

If matrix B results from matrix A by interchanging two rows(columns) of A, then det(B) = − det(A).

If matrix B is obtained from matrix A by multiplying a row(column) of A by a real number c, then det(B) = c det(A).

Determinants

Some Properties The determinants of a matrix and its transpose are equal, that is, det(AT ) = det(A) where the transpose of A is obtained by interchanging the rows and columns of A.

If matrix B results from matrix A by interchanging two rows(columns) of A, then det(B) = − det(A).

If matrix B is obtained from matrix A by multiplying a row(column) of A by a real number c, then det(B) = c det(A).

If B = [bij ] is obtained from A = [aij ] by adding to each element of the rth row(column) of A a constant c times the corresponding element of the sth row(column) r 6= s of A, then det(B) = det(A).

Determinants

More Properties

If two rows(columns) A are equal, then det(A) = 0.

Determinants

More Properties

If two rows(columns) A are equal, then det(A) = 0.

If a row(column) of A consists entirely of zeros, then det(A) = 0.

Determinants

More Properties

If two rows(columns) A are equal, then det(A) = 0.

If a row(column) of A consists entirely of zeros, then det(A) = 0.

The determinant of a diagonal matrix is the product of the entries on its main diagonal.

Determinants

More Properties

If two rows(columns) A are equal, then det(A) = 0.

If a row(column) of A consists entirely of zeros, then det(A) = 0.

The determinant of a diagonal matrix is the product of the entries on its main diagonal. The determinant of an upper(lower) triangular matrix is the product of the entries on its main diagonal.

Determinants

More Properties

If two rows(columns) A are equal, then det(A) = 0.

If a row(column) of A consists entirely of zeros, then det(A) = 0.

The determinant of a diagonal matrix is the product of the entries on its main diagonal. The determinant of an upper(lower) triangular matrix is the product of the entries on its main diagonal.

The determinant of a product of two matrices is the product of their determinants, det(AB) = det(A) det(B).

Determinants

More Properties

If two rows(columns) A are equal, then det(A) = 0.

If a row(column) of A consists entirely of zeros, then det(A) = 0.

The determinant of a diagonal matrix is the product of the entries on its main diagonal. The determinant of an upper(lower) triangular matrix is the product of the entries on its main diagonal.

The determinant of a product of two matrices is the product of their determinants, det(AB) = det(A) det(B).

If A is nonsingular, then det(A) 6= 0 and det(A−1 ) =

1 . det(A)

Determinants

The Cofactor

Definition Let A = [aij ] be an n × n matrix.

Let Mij be the (n − 1) × (n − 1) submatrix of A obtained by deleting the ith row and jth column of A. The determinant det(Mij ) is called the minor of aij .

Determinants

The Cofactor

Definition Let A = [aij ] be an n × n matrix.

Let Mij be the (n − 1) × (n − 1) submatrix of A obtained by deleting the ith row and jth column of A. The determinant det(Mij ) is called the minor of aij .

The cofactor Aij of aij is defined as Aij = (−1)i+j det(Mij ).

Determinants

Cofactor Expansion

Let A = [aij ] be an n × n matrix. Then for each 1 ≤ i ≤ n, det(A) = ai1 Ai1 + ai2 Ai2 + · · · + ain Ain (expansion of det(A) along the ith row); and for each 1 ≤ j ≤ n, det(A) = a1j A1j + a2j A2j + · · · + anj Anj (expansion of det(A) along the jth column);

Determinants

Cofactor Expansion

Let A = [aij ] be an n × n matrix. Then for each 1 ≤ i ≤ n, det(A) = ai1 Ai1 + ai2 Ai2 + · · · + ain Ain (expansion of det(A) along the ith row); and for each 1 ≤ j ≤ n, det(A) = a1j A1j + a2j A2j + · · · + anj Anj (expansion of det(A) along the jth column);

Example

1 4 3 1

−2 4 0

1 0 2

0 2 −5

4 0 2

Determinants

The Adjoint

Definition Let A = [aij ] be an n × n matrix. The n × n matrix adj A, called the adjoint of A, is the matrix whose i, jth element is the cofactor Aji of aji . Thus   A11 A21 · · · An1  A12 A22 · · · An2   adj A =   : : :  A1n A2n · · · Ann

Determinants

The Adjoint

Definition Let A = [aij ] be an n × n matrix. The n × n matrix adj A, called the adjoint of A, is the matrix whose i, jth element is the cofactor Aji of aji . Thus   A11 A21 · · · An1  A12 A22 · · · An2   adj A =   : : :  A1n A2n · · · Ann

Example 

1 A= 0 −3

5 1 2

 2 0  0

Determinants

The Adjoint

Definition Let A = [aij ] be an n × n matrix. The n × n matrix adj A, called the adjoint of A, is the matrix whose i, jth element is the cofactor Aji of aji . Thus   A11 A21 · · · An1  A12 A22 · · · An2   adj A =   : : :  A1n A2n · · · Ann

If A = [aij ] is an n × n matrix, then A(adj A) = (adj A)A = det(A)In .

Determinants

The Inverse

If A is an n Ă— n matrix and det(A) 6= 0, then A−1 =

1 (adj A). det(A)

Determinants

The Inverse

If A is an n × n matrix and det(A) 6= 0, then A−1 =

1 (adj A). det(A)

Example 

1 A= 0 −3

5 1 2

 2 0  0

Determinants

The Inverse

If A is an n Ă— n matrix and det(A) 6= 0, then A−1 =

1 (adj A). det(A)

A matrix A is nonsingular if and only if det(A) 6= 0.

Determinants

Cramer’s Rule Consider the linear system of n equations in n unknowns, a11 x1 + a12 x2 + · · · + a1n xn = b1 a21 x1 + a22 x2 + · · · + a2n xn = b2 : : : : an1 x1 + an2 x2 + · · · + ann xn = bn This linear  system a11  a21 where A =   : an1

can be written in matrix formas Ax = b a12 · · · a1n x1 b1  x2   b2 a22 · · · a2n  , x =     : , b =  : : :  an2 · · · ann xn bn

   

If det(A) 6= 0, then the system has the unique solution x1 =

det(A1 ) , det(A)

x2 =

det(A2 ) , det(A)

··· ,

xn =

det(An ) det(A)

where Ai is the matrix obtained from A by replacing the ith column of A by b.

Determinants

Exercises Solve the ff. linear systems: 1. −2x1 x1 −2x1 2. 3x x 2x

+ − +

+ + − 2y y y

3x2 2x2 x2 − − −

− x3 − x3 + x3 z z 2z

= 1 = 4 = −3

= −1 = 0 = 3

Determinants

List of Nonsingular Equivalences Suppose A is an n Ă— n matrix.

Then the following statements are equivalent: 1. A is nonsingular.

2. A is row equivalent to In .

3. Ax = 0 has only the trivial solution.

4. The linear system Ax = b has a unique solution for every n Ă— 1 matrix b.

5. det(A) 6= 0.

Determinants

For Self-Study

Determinants

Scalar Multiplication

Definition If A = [aij ] is an m × n matrix and r is a real number, then the scalar multiple of A by r, rA, is the m × n matrix B = [bij ], where bij = r · aij , 1 ≤ i ≤ m, 1 ≤ j ≤ n.

That is, B is obtained by multiplying each element of A by r.

Determinants

Properties of Scalar Multiplication

Theorem If r and s are real numbers and A and B are matrices, then (a) r(sA) = (rs)A (b) (r + s)A = rA + sA (c) r(A + B) = rA + rB (d) A(rB) = r(AB) = (rA)B

Determinants

Properties of Transpose

Theorem If r is a real number and A and B are matrices, then (a) (AT )T = A (b) (A + B)T = AT + B T (c) (AB)T = B T AT (d) (rA)T = rAT

Determinants

Properties of Transpose

A matrix A = [aij ] is called symmetric if AT = A. That is, A is symmetric if it is a square matrix for which aij = aji .

If matrix A is symmetric, then the elements of A are symmetric with respect to the main diagonal of A.

Determinants

Questions?

See you next meeting!