Matrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices
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1 Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016
2 Table of contents Matrix Algebra Determinant Inverse Matrix
3 Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated by the number of rows and the number of its columns. A matrix with k rows and n columns is called a k n ( k by n ) matrix. The number in row i and column j is called the (i, j)th entry and it is denoted by a ij. The matrix A with entries a ij with i = 1,..., k and j = 1,...n is a k n matrix is representing as follows A = a 11 a 1n.. a k1 a kn Matrices can be added, subtracted, multiplied and even divided if their sizes respect some conditions.
4 Addition Two matrices can be added only if they have the same size, i.e. the same number of rows and columns. Their sum is a new matrix of the same size as the two matrices being added. Let A and B be two k n matrices with entries a ij and b ij for i = 1,..., k and j = 1,..., n, respectively. Their sum A + B is a k n matrix with entries a ij + b ij for i = 1,..., k and j = 1,..., n.
5 Addition (2) Explicitly A + B = a 11 a 1n. a ij. a k1 a kn + b 11 b 1n. b ij. b k1 b kn = a 11 + b 11 a 1n + b 1n. a ij + b ij. a k1 + b k1 a kn + b kn
6 Addition: Example ( Let A = matrices. ) and B = ( ) two 2 3 Their sum is a 2 3 matrix ( ) ( A+B = ) = ( )
7 Addition The matrix 0 is a special matrix whose entries are all zero. It is an additive identity since A + 0 = A for all matrices A: a 11 a 1n 0 0 a 11 a 1n. a ij =. a ij. a k1 a kn 0 0 a k1 a kn
8 Subtraction Subtraction, as addition, is possible only if the two matrices have the same size. Writing A B is a shorthand for A + ( B) where B is what one adds to B to get 0 (as opposite for numbers). A B = a 11 a 1n. a ij. a k1 a kn b 11 b 1n. b ij. b k1 b kn = a 11 b 11 a 1n b 1n. a ij b ij. a k1 b k1 a kn b kn
9 Scalar multiplication Matrices can be multiplied by real numbers called scalar, the result is a matrix with the same size with all the entries multiplied by the scalar. Explicitly, let A be a k n matrices and γ a scalar (γ R) a 11 a 1n γa = γ. a ij. a k1 a kn = γa 11 γa 1n. γa ij. γa k1 γa kn
10 Exercises Given A = and β = 3. Solve the following expression 1. βa + α(b A) 2. B αb + A 3. (α + β)(a + B), B /2 2 1/2 2, α = 2
11 Matrix Multiplication In matrix multiplication sizes and order in which matrices are multiplied matter! We can define a matrix product of A and B, AB, only if the number of columns of A equals the number of rows of B. Let A be a k m matrix and B a m n matrix, their product AB exists and it is a matrix k n whose (i, j) th entry is given by the inner product of the ith row of A and jth column of B (AB) ij = (a i1 a i2 a im ) b 1j b 2j. b mj = a i1b 1j a i2 b 2j a im b mj.
12 Matrix Multiplication (2) In other words, the (i, j) th entry is given by m a ih b hj. h=1 Note that if AB exists, BA is not equal (this operation is not commutative) and even does not exist! Indeed, if k n, BA cannot be calculated.
13 Matrix Multiplication: Examples 1. Calculate AB and say if BA exists A = and B = Calculate CD anddc and verify that CD DC. 1 1 ( ) C = and D =
14 Identity Matrix The n n matrix I = , is called the identity matrix and is such that a ii = 1 for all i and a ij = 0 for all i j. This matrix has the property that for any m n matrix A AI = A and for any n l matrix B IB = B.
15 Law of Matrix Algebra Associative law Commutative law for addition Distributive law (A + B) + C = A + (B + C) (AB)C = A(BC) A + B = B + A A(B + C) = AB + AC (A + B)C = AC + BC Recall that matrix multiplication is not commutative!
16 Transpose The transpose of a k n matrix is a n k matrix obtained by interchanging the rows and the columns of A. This matrix is denoted by A T. Thus the (i, j)th entry of A becomes the (j, i)th entry of A T. For example a 11 a 12 a 21 a 22 a 31 a 32 T ( ) a11 a = 21 a 31 a 12 a 22 a 32
17 Laws for transpose matrices Let A and B be k n matrices and γ a scalar, the following rules hold (A + B) T = A T + B T, (A B) T = A T B T, (A T ) T = A, (γa) T = γa T
18 Transpose of matrices product Let A be k m matrix and B be m n matrix. Then, (AB) T = B T A T. This can be easily proved ((AB) T ) ij = (AB) ji = h A jhb hi = h (AT ) hj (B T ) ih = h (BT ) ih (A T ) hj = (B T A T ) ij
19 Exercises Using matrices A = C = ( ( calculate (where possible) 1. A + B 2D 2. (2A 3B) T D 3. (AD) T C + B ) ( 1 2, B = 1 1 ) ( 2 3, D = 1 4 ), ),
20 Here a list of special matrices: A matrix is called square if it has the same number of columns and rows (k = n) A matrix is a column matrix if it has only one column. For example a b c A matrix is a row matrix if it has only one row. For example ( e f d )
21 (2) Special matrices list follows: A diagonal matrix is a square matrix in which all nondiagonal entries are zeros. For example a b c A upper-triangular matrix is a (square) matrix in which all entries below the diagonal are zeros. For example
22 Special Matrix (3) list follows: A lower-triangular matrix is a (square) matrix in which all entries above the diagonal are zeros. For example A square matrix B is idempotent if BB = B.
23 Symmetric Matrix A square matrix is symmetric if A = A T or similarly if a ij = a ji for all i, j. For example Example: The variance covariance matrix is a symmetric matrix..
24 Exercises 1. ( Find real) numbers ( a, ) b and ( x such ) ( that a b a b x x 0 ) = ( A square matrix B is anti-symmetric (or skew-symmetric) is B = B T. Show that if A is any square matrix, A 1 = 1 2 (A + AT ) is symmetric and A 2 = 1 2 (A AT ) is anti-symmetric. Verify that A = A 1 + A If P and Q are n n matrices with PQ QP = P, prove that P 2 Q QP 2 = 2P 2 and P 3 Q QP 3 = 3P 3. Then use induction to prove that P k Q QP k = kp k for k = 1, 2,... ).
25 Square Matrices Within the class of n n matrices, denoted by M n, all the arithmetic operations are possibile and the result is a n n matrix. The identity matrix I is such that AI = IA = A. Since we have the product, can we divide square matrices? We must introduce the idea of inverse as for numbers. Definition Let A be a matrix in M n. The matrix B M n is an inverse for A if AB = BA = I. If the matrix B exists we say that A is invertible.
26 Inverse matrix is unique Theorem An n n matrix can have at most one inverse Proof. Suppose that B and C are both inverse of A. Then we have B = BI = B(AC) = (BA)C = IC = C. If a matrix A M n is invertible we denote its unique inverse A 1.
27 Exercises ( ) 2 1 Verify that A = is invertible and 1 1 ( ) 1 1 A 1 = Verify that B = is invertible and B =
28 Inverse matrix of a 2 2 matrix Given a 2 2 matrix ( ) a b A = c d the inverse exists if ad bc 0 and ( A 1 1 d b = ad bc c a )
29 Theorem Let A and B be square invertible matrices. Then (A 1 ) 1 = A; (A T ) 1 = (A 1 ) T ; AB is invertible and (AB) 1 = B 1 A 1. Theorem If A is invertible: (A m ) is invertible for any integer m and (A m ) 1 = (A 1 ) m ; for any integer r and s A r A s = A r+s ; for any scalar γ, γa is invertible and (γa) 1 = γ 1 A 1.
30 Exercises ( ) Given A = find (if it exists) A Show that the inverse of a 2 2 symmetric matrix is symmetric. 3. What is the inverse of an n n diagonal matrix? When does it exist? 4. Show that (A + B) 1, if it exists, is generally not A 1 + B 1.
31 Determinant Inverse Matrix Defining the determinant The determinant of a matrix can be defined inductively. (n = 1) A 1 1 matrix is a scalar a. The inverse of such a matrix exists if a 0 hence we define det(a) = a ( a11 a (n = 2) A matrix 2 2 A = 12 a 21 a 22 a 11 a 22 a 12 a 21 0 hence we define ) is invertible if det(a) = a 11 a 22 a 12 a 21 which equals det(a) = a 11 det(a 22 ) a 12 det(a 21 ).
32 Determinant Inverse Matrix Definition Let A be an n n matrix. Let A ij be the (n 1) (n 1) submatrix obtained by deleting row i and column j from A. Then the scalar M ij = det(a ij ) is called the (i, j)th minor of A and the scalar is called the (i, j)th cofactor. C ij = ( 1) i+j M ij
33 Determinant Inverse Matrix The determinant of a 2 2 matrix The determinant of a 2 2 matrix det(a) = a 11 a 22 a 12 a 21 can be seen det(a) = a 11 det(a 22 ) a 12 det(a 21 ) = a 11 M 11 a 12 M 12 = a 11 C 11 + a 12 C 12
34 Determinant Inverse Matrix The determinant of a 3 3 matrix The determinant of a 3 3 matrix a 11 a 12 a 13 det a 21 a 22 a 23 = a 11 C 11 + a 12 C 12 + a 13 C 13 a 31 a 32 a 33 = a 11 M 11 a 12 M 12 + a 13 M 13 ( a22 a = a 11 det 23 a 32 a 33 ) ( a21 a a 12 det 23 a 31 a 33 ) ( a21 a + a 13 det 22 a 31 a 32 )
35 Determinant Inverse Matrix The determinant of a n n matrix Definition The determinant of a n n matrix det(a) = a 11 C 11 + a 12 C a 1n C 1n = a 11 M 11 a 12 M ( 1) n+1 a 1n M 1n Notation In referring to determinant of matrix A we write det(a), A or a 11 a 1n a 11 a 1n. a ij. or det. a ij. a n1 a nn a n1 a nn
36 Determinant Inverse Matrix Exercises Compute the determinants of the following matrices A = B = C = D = ( )
37 Determinant Inverse Matrix Theorem The determinant of a lower-triangular, an upper-triangular or a diagonal matrix is simply the product of its diagonal entries. Proof. Easy to see when 2 2. For a upper-triangular 3 3: det a 11 a 12 a 13 0 a 22 a a 33 = a 11 C 11 + a 12 C 12 + a 13 C 13 a = a 22 a a 33 a 12 0 a 23 0 a 33 0 a +a = a 11a 22 a 33.
38 Determinant Inverse Matrix Theorem Let A be a square matrix. Then det(a T ) = det(a) det(ab) = det(a) det(b) det(a + B) det(a) + det(b) det(γa) = γ n det(a) for γ scalar
39 Determinant Inverse Matrix Adjoint matrix For n n matrix A we denote by C ij the (i, j)th cofactor of A, that is the ( 1) i+j times the determinant of the submatrix obtained deleting row i and columns j from A. Definition The n n matrix whose (i, j)th entry is C ji is called the adjoint of A and is denoted by adja. Note that we can alternatively say that the adjoint matrix is the transpose of the matrix having as entries the cofactors of A.
40 Determinant Inverse Matrix Inverse matrix Theorem Let A be an n n invertible matrix. Then, A 1 = 1 det(a) adj(a). Note that a matrix is invertible if and only if its determinant is different from zero. Note that AA 1 = I thus det(aa 1 ) = det(i ) = 1, since det(aa 1 ) = det(a) det(a 1 ) we derive that det(a 1 ) = 1 det(a).
41 Determinant Inverse Matrix Exercises Find the inverse (if it exists) of the following matrices A = B = C =
42 Determinant Inverse Matrix Exercises Exericise 1 Let A t = 1 0 t 2 1 t and B = 1. For what values of t does A t have an inverse? 2. Find a matrix X such that B + XA 1 1 = A 1 1. Exericise 2 Solve the equation 1 x x x =
43 Determinant Inverse Matrix Trace Definition The trace of a square matrix is the sum of its diagonal entries: trace(a) = a 11 + a a nn.
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