Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Note 8 : MATRIX ALGEBRA - Department of Computer Science
Document related concepts
no text concepts found
Transcript
Note 8 : MATRIX ALGEBRA 01 Note 8 : MATRIX ALGEBRA CS131: Mathematics for Computer Science II Part II, Linear Algebra and Matrices Note 8 : MATRIX ALGEBRA Matrices A matrix is a rectangular array of objects. 02 Note 8 : MATRIX ALGEBRA Matrices A matrix is a rectangular array of objects. We shall usually consider only matrices of real numbers but more generally the elements can be objects such as complex numbers, rational numbers, integers, integers modulo n for some n ∈ N, or even other matrices. 03 Note 8 : MATRIX ALGEBRA Matrices A matrix is a rectangular array of objects. We shall usually consider only matrices of real numbers but more generally the elements can be objects such as complex numbers, rational numbers, integers, integers modulo n for some n ∈ N, or even other matrices. 1 5 −3 2 A 2 × 2 matrix 04 Note 8 : MATRIX ALGEBRA 05 Matrices A matrix is a rectangular array of objects. We shall usually consider only matrices of real numbers but more generally the elements can be objects such as complex numbers, rational numbers, integers, integers modulo n for some n ∈ N, or even other matrices. 1 5 −3 2 A 2 × 2 matrix 3π 0 − 23 e √ 2 π 3 A 3 × 2 matrix Note 8 : MATRIX ALGEBRA 06 Matrices A matrix is a rectangular array of objects. We shall usually consider only matrices of real numbers but more generally the elements can be objects such as complex numbers, rational numbers, integers, integers modulo n for some n ∈ N, or even other matrices. 1 5 −3 2 A 2 × 2 matrix 3π 0 − 23 e √ 2 π 3 A 3 × 2 matrix 2 + 3i 1−i 3 − 7i i −4 6 − 2i A 2 × 3 matrix Note 8 : MATRIX ALGEBRA The Order of a Matrix A matrix A of order m × n is an array of numbers arranged in m rows and n columns and usually written as a a · · · · · · a1n 11 12 a a · · · · · · a 21 22 2n A= . or A = [aij ]m×n . . .. .. am1 · · · · · · · · · amn 07 Note 8 : MATRIX ALGEBRA The Order of a Matrix A matrix A of order m × n is an array of numbers arranged in m rows and n columns and usually written as a a · · · · · · a1n 11 12 a a · · · · · · a 21 22 2n A= . or A = [aij ]m×n . . .. .. am1 · · · · · · · · · amn The numbers aij are called the elements (or entries) of A. 08 Note 8 : MATRIX ALGEBRA The Order of a Matrix A matrix A of order m × n is an array of numbers arranged in m rows and n columns and usually written as a a · · · · · · a1n 11 12 a a · · · · · · a 21 22 2n A= . or A = [aij ]m×n . . .. .. am1 · · · · · · · · · amn The numbers aij are called the elements (or entries) of A. Element aij is in the i th row and j th column. 09 Note 8 : MATRIX ALGEBRA 10 Column Vectors and Row Vectors • A column matrix or column vector is a m × 1 matrix. a 11 a21 Column vector: .. . am1 Note 8 : MATRIX ALGEBRA 11 Column Vectors and Row Vectors • A column matrix or column vector is a m × 1 matrix. a 11 a21 Column vector: .. . am1 • A row matrix or row vector is a 1 × n matrix. Row vector: [b11 b12 · · · b1n ] Note 8 : MATRIX ALGEBRA 12 Column Vectors and Row Vectors • A column matrix or column vector is a m × 1 matrix. a 11 a21 Column vector: .. . am1 • A row matrix or row vector is a 1 × n matrix. Row vector: [b11 b12 · · · b1n ] • It is often useful to think of members of Rn as either row vectors or column vectors in different contexts. Note 8 : MATRIX ALGEBRA Equality of Matrices Two matrices A and B are equal if they have the same order and all the corresponding elements are equal 13 Note 8 : MATRIX ALGEBRA Equality of Matrices Two matrices A and B are equal if they have the same order and all the corresponding elements are equal i.e. A = [aij ]m×n and B = [bij ]m×n for some m and n and aij = bij for every i and j . 14 Note 8 : MATRIX ALGEBRA Addition of Matrices • The sum A + B of two matrices A and B is defined only when A and B have the same order. • The sum is obtained by adding corresponding elements. • Thus A+B = [aij + bij ]m×n if A = [aij ]m×n and B = [bij ]m×n . 15 Note 8 : MATRIX ALGEBRA Addition of Matrices • The sum A + B of two matrices A and B is defined only when A and B have the same order. • The sum is obtained by adding corresponding elements. • Thus A+B = [aij + bij ]m×n if A = [aij ]m×n and B = [bij ]m×n . Scalar Multiplication • In scalar multiplication every element in the matrix is multiplied by the scalar. • Thus λ A = [λ aij ]m×n if A = [aij ]m×n and λ ∈ R. 16 Note 8 : MATRIX ALGEBRA 17 The Zero Matrix • The zero matrix of order m × n is the m × n matrix whose elements are all 0. • The zero matrix of order m × n is denoted by O or Om×n . Note 8 : MATRIX ALGEBRA 18 The Zero Matrix • The zero matrix of order m × n is the m × n matrix whose elements are all 0. • The zero matrix of order m × n is denoted by O or Om×n . The Negative of Matrix A • The negative of matrix A = [aij ]m×n is the matrix −A = [−aij ]m×n . Note 8 : MATRIX ALGEBRA Properties of Addition and Scalar Multiplication For any m × n matrices A and B and any λ, µ ∈ R: 19 Note 8 : MATRIX ALGEBRA 20 Properties of Addition and Scalar Multiplication For any m × n matrices A and B and any λ, µ ∈ R: (1) A + (B + C ) = (A + B ) + C [associativity of addition] Note 8 : MATRIX ALGEBRA 21 Properties of Addition and Scalar Multiplication For any m × n matrices A and B and any λ, µ ∈ R: (1) A + (B + C ) = (A + B ) + C (2) A + O = A = O + A [associativity of addition] [additive identity] Note 8 : MATRIX ALGEBRA 22 Properties of Addition and Scalar Multiplication For any m × n matrices A and B and any λ, µ ∈ R: (1) A + (B + C ) = (A + B ) + C (2) A + O = A = O + A (3) A + (−A) = O = (−A) + A [associativity of addition] [additive identity] [additive inverse] Note 8 : MATRIX ALGEBRA 23 Properties of Addition and Scalar Multiplication For any m × n matrices A and B and any λ, µ ∈ R: (1) A + (B + C ) = (A + B ) + C (2) A + O = A = O + A (3) A + (−A) = O = (−A) + A (4) A + B = B + A [associativity of addition] [additive identity] [additive inverse] [commutativity of addition] Note 8 : MATRIX ALGEBRA 24 Properties of Addition and Scalar Multiplication For any m × n matrices A and B and any λ, µ ∈ R: (1) A + (B + C ) = (A + B ) + C (2) A + O = A = O + A (3) A + (−A) = O = (−A) + A (4) A + B = B + A (5) (λ + µ)A = λ A + µ A [associativity of addition] [additive identity] [additive inverse] [commutativity of addition] [distributivity] Note 8 : MATRIX ALGEBRA 25 Properties of Addition and Scalar Multiplication For any m × n matrices A and B and any λ, µ ∈ R: (1) A + (B + C ) = (A + B ) + C (2) A + O = A = O + A (3) A + (−A) = O = (−A) + A (4) A + B = B + A [associativity of addition] [additive identity] [additive inverse] [commutativity of addition] (5) (λ + µ)A = λ A + µ A [distributivity] (6) λ(A + B ) = λ A + λ B [distributivity] Note 8 : MATRIX ALGEBRA 26 Properties of Addition and Scalar Multiplication For any m × n matrices A and B and any λ, µ ∈ R: (1) A + (B + C ) = (A + B ) + C (2) A + O = A = O + A (3) A + (−A) = O = (−A) + A (4) A + B = B + A [associativity of addition] [additive identity] [additive inverse] [commutativity of addition] (5) (λ + µ)A = λ A + µ A [distributivity] (6) λ(A + B ) = λ A + λ B [distributivity] (7) λ(µ A) = (λ µ)A [associativity] Note 8 : MATRIX ALGEBRA Multiplying Matrices The product of two matrices is defined only when the number of columns in the first is the same as the number of rows in the second. 27 Note 8 : MATRIX ALGEBRA Multiplying Matrices The product of two matrices is defined only when the number of columns in the first is the same as the number of rows in the second. The product AB = [cij ]m×n of A = [aij ]m×p and B = [bij ]p×n is the m × n matrix where • The ij th element of AB is the scalar product of the i th row vector of A with the j th column vector of B . 28 Note 8 : MATRIX ALGEBRA Multiplying Matrices The product of two matrices is defined only when the number of columns in the first is the same as the number of rows in the second. The product AB = [cij ]m×n of A = [aij ]m×p and B = [bij ]p×n is the m × n matrix where • The ij th element of AB is the scalar product of the i th row vector of A with the j th column vector of B . Pp • i.e. cij = r =1 air brj = (ai1 , ai2 , . . . , aip ).(b1j , b2j , . . . , bpj ). 29 Note 8 : MATRIX ALGEBRA 30 Multiplying Matrices The product of two matrices is defined only when the number of columns in the first is the same as the number of rows in the second. The product AB = [cij ]m×n of A = [aij ]m×p and B = [bij ]p×n is the m × n matrix where • The ij th element of AB is the scalar product of the i th row vector of A with the j th column vector of B . Pp • i.e. cij = r =1 air brj = (ai1 , ai2 , . . . , aip ).(b1j , b2j , . . . , bpj ). a11 · · · · · · a1p . . . . . . ai1 .. . am1 b11 · · · .. . ai2 · · · aip .. . . . . bp1 · · · · · · · · · amp b1j ··· b2j . . . bpj ··· b1n . . . . . . bpn c11 · · · · · · · · · c1n . .. . . . = cij . . . . . . cm1 · · · · · · · · · cmn Note 8 : MATRIX ALGEBRA 31 Examples " (i) 1 2 0 −3 2 0 −1 −5 0 1 1 −3 1 4 3 0 2 −1 # Note 8 : MATRIX ALGEBRA 32 Examples " (i) # " 1 2 0 −3 2 0 −1 −1 4 −2 −5 −5 0 1 1 = −3 1 4 4 −6 9 6 3 0 2 −1 # Note 8 : MATRIX ALGEBRA 33 Examples # " 1 2 0 −3 2 0 −1 −1 4 −2 −5 (i) −5 0 1 1 = −3 1 4 4 −6 9 6 3 0 2 −1 h i −1 (ii) 2 1 −3 −2 0 " # Note 8 : MATRIX ALGEBRA 34 Examples # " 1 2 0 −3 2 0 −1 −1 4 −2 −5 (i) −5 0 1 1 = −3 1 4 4 −6 9 6 3 0 2 −1 h i −1 h i (ii) 2 1 −3 −2 = −4 0 " # Note 8 : MATRIX ALGEBRA 35 Examples # " 1 2 0 −3 2 0 −1 −1 4 −2 −5 (i) −5 0 1 1 = −3 1 4 4 −6 9 6 3 0 2 −1 h i −1 h i (ii) 2 1 −3 −2 = −4 0 1 h i (iii) −1 1 2 0 3 0 " # Note 8 : MATRIX ALGEBRA 36 Examples # " 1 2 0 −3 2 0 −1 −1 4 −2 −5 (i) −5 0 1 1 = −3 1 4 4 −6 9 6 3 0 2 −1 h i −1 h i (ii) 2 1 −3 −2 = −4 0 1 h 1 20 3 i (iii) −1 1 2 0 3 = −1 −2 0 −3 0 0 00 0 " # Note 8 : MATRIX ALGEBRA 37 Examples (i) (ii) (iii) (iv) # " 1 2 0 −3 2 0 −1 −1 4 −2 −5 −5 0 1 1 = −3 1 4 4 −6 9 6 3 0 2 −1 h i −1 h i 2 1 −3 −2 = −4 0 1 h 1 20 3 i −1 1 2 0 3 = −1 −2 0 −3 0 0 00 0 " #" # a11 a12 x1 a21 a22 x2 " # Note 8 : MATRIX ALGEBRA 38 Examples (i) (ii) (iii) (iv) # " 1 2 0 −3 2 0 −1 −1 4 −2 −5 −5 0 1 1 = −3 1 4 4 −6 9 6 3 0 2 −1 h i −1 h i 2 1 −3 −2 = −4 0 1 h 1 20 3 i −1 1 2 0 3 = −1 −2 0 −3 0 0 00 0 " #" # " # a11 a12 x1 a11 x1 + a12 x2 = . a21 a22 x2 a21 x1 + a22 x2 " # Note 8 : MATRIX ALGEBRA Square Matrices • A square matrix is one with the same number of rows as columns i.e. of the form A = [aij ]n×n for some n. 39 Note 8 : MATRIX ALGEBRA Square Matrices • A square matrix is one with the same number of rows as columns i.e. of the form A = [aij ]n×n for some n. – We call A = [aij ]n×n a square matrix of order n. 40 Note 8 : MATRIX ALGEBRA Square Matrices • A square matrix is one with the same number of rows as columns i.e. of the form A = [aij ]n×n for some n. – We call A = [aij ]n×n a square matrix of order n. • Products AB and BA are both defined if A and B are square matrices of order n. 41 Note 8 : MATRIX ALGEBRA Square Matrices • A square matrix is one with the same number of rows as columns i.e. of the form A = [aij ]n×n for some n. – We call A = [aij ]n×n a square matrix of order n. • Products AB and BA are both defined if A and B are square matrices of order n. • Matrix multiplication is not commutative – i.e. AB and BA are not necessarily equal. 42 Note 8 : MATRIX ALGEBRA 43 Square Matrices • A square matrix is one with the same number of rows as columns i.e. of the form A = [aij ]n×n for some n. – We call A = [aij ]n×n a square matrix of order n. • Products AB and BA are both defined if A and B are square matrices of order n. • Matrix multiplication is not commutative – i.e. AB and BA are not necessarily equal. " #" # " # – 01 00 10 e.g. = but 00 10 00 " 00 10 #" # " # 01 00 = . 00 01 Note 8 : MATRIX ALGEBRA 44 Square Matrices • A square matrix is one with the same number of rows as columns i.e. of the form A = [aij ]n×n for some n. – We call A = [aij ]n×n a square matrix of order n. • Products AB and BA are both defined if A and B are square matrices of order n. • Matrix multiplication is not commutative – i.e. AB and BA are not necessarily equal. " #" # " # – 01 00 10 e.g. = but 00 10 00 " 00 10 #" # " # 01 00 = . 00 01 • If A is a square matrix then the products AA, AAA, AAAA, . . . are all defined and are denoted by A2 , A3 , A4 , . . .. Note 8 : MATRIX ALGEBRA Diagonal Matrices • In a square matrix A = [aij ]n×n , the diagonal elements are those of the form aii for some i . 45 Note 8 : MATRIX ALGEBRA Diagonal Matrices • In a square matrix A = [aij ]n×n , the diagonal elements are those of the form aii for some i . • A diagonal matrix is a square matrix whose non-diagonal elements are all zero. 46 Note 8 : MATRIX ALGEBRA 47 Diagonal Matrices • In a square matrix A = [aij ]n×n , the diagonal elements are those of the form aii for some i . • A diagonal matrix is a square matrix whose zero. a11 0 0 e.g. 0 a22 0 0 0 a33 non-diagonal elements are all other notations: diag[a11 , a22 , a33 ]. diag[aii ] Note 8 : MATRIX ALGEBRA Identity Matrices. • The identity matrix of order n is the n × n diagonal matrix whose diagonal elements are all 1. 48 Note 8 : MATRIX ALGEBRA 49 Identity Matrices. • The identity matrix of order n is the n × n diagonal matrix whose diagonal elements are all 1. • It is denoted by I or In . – e.g. I1 = [1], I2 = 10 01 , 100 I3 = 0 1 0 . 001 Note 8 : MATRIX ALGEBRA 50 Identity Matrices. • The identity matrix of order n is the n × n diagonal matrix whose diagonal elements are all 1. • It is denoted by I or In . – e.g. I1 = [1], I2 = 10 01 , 100 I3 = 0 1 0 . 001 • If A is a square matrix then A0 denotes the identity matrix I of the same order. Note 8 : MATRIX ALGEBRA 51 Identity Matrices. • The identity matrix of order n is the n × n diagonal matrix whose diagonal elements are all 1. • It is denoted by I or In . – e.g. I1 = [1], I2 = 10 01 , 100 I3 = 0 1 0 . 001 • If A is a square matrix then A0 denotes the identity matrix I of the same order. The functions exp A, sin A and cos A can also be defined. Note 8 : MATRIX ALGEBRA 52 Properties of Matrix Multiplication The following identities hold whenever the products on the left-hand side (or right-hand side) exist: (1) (AB )C = A(BC ) associativity of × Note 8 : MATRIX ALGEBRA 53 Properties of Matrix Multiplication The following identities hold whenever the products on the left-hand side (or right-hand side) exist: (1) (AB )C = A(BC ) (2a) A(B + C ) = AB + AC (2b) (A + B )C = AC + BC associativity of × distributivity of × over + distributivity of + over × Note 8 : MATRIX ALGEBRA 54 Properties of Matrix Multiplication The following identities hold whenever the products on the left-hand side (or right-hand side) exist: (1) (AB )C = A(BC ) (2a) A(B + C ) = AB + AC (2b) (A + B )C = AC + BC (3) IA = A = AI associativity of × distributivity of × over + distributivity of + over × multiplicative identity Note 8 : MATRIX ALGEBRA 55 Properties of Matrix Multiplication The following identities hold whenever the products on the left-hand side (or right-hand side) exist: (1) (AB )C = A(BC ) (2a) A(B + C ) = AB + AC (2b) (A + B )C = AC + BC (3) IA = A = AI (4) OA = O = AO associativity of × distributivity of × over + distributivity of + over × multiplicative identity additive identity Note 8 : MATRIX ALGEBRA 56 Properties of Matrix Multiplication The following identities hold whenever the products on the left-hand side (or right-hand side) exist: (1) (AB )C = A(BC ) (2a) A(B + C ) = AB + AC (2b) (A + B )C = AC + BC (3) IA = A = AI (4) OA = O = AO (5a) Ap Aq = Ap+q = Aq Ap (5b) (Ap )q = Apq associativity of × distributivity of × over + distributivity of + over × multiplicative identity additive identity indices indices Note 8 : MATRIX ALGEBRA Transpose • The transpose AT of matrix A is obtained by interchanging the rows and columns. 57 Note 8 : MATRIX ALGEBRA Transpose • The transpose AT of matrix A is obtained by interchanging the rows and columns. • Thus if A = [aij ]m×n then AT = [a 0ij ]n×m where a 0ij = aji . 58 Note 8 : MATRIX ALGEBRA 59 Transpose • The transpose AT of matrix A is obtained by interchanging the rows and columns. • Thus if A = [aij ]m×n then AT = [a 0ij ]n×m where a 0ij = aji . Examples 352 234 A= , B = 1 0 1 , C = [a1 a2 . . . an ] 105 216 " If # Note 8 : MATRIX ALGEBRA 60 Transpose • The transpose AT of matrix A is obtained by interchanging the rows and columns. • Thus if A = [aij ]m×n then AT = [a 0ij ]n×m where a 0ij = aji . Examples 352 234 A= , B = 1 0 1 , C = [a1 a2 . . . an ] 105 216 a1 21 312 a2 T T T A = 3 0, B = 5 0 1, C = .. . . 45 216 an " If then # Note 8 : MATRIX ALGEBRA 61 Properties of Transposes (1) (AT )T = A Note 8 : MATRIX ALGEBRA 62 Properties of Transposes (1) (AT )T = A (2) (A + B )T = AT + B T when A + B exists Note 8 : MATRIX ALGEBRA 63 Properties of Transposes (1) (AT )T = A (2) (A + B )T = AT + B T when A + B exists (3) (λ A)T = λ AT for any λ ∈ R Note 8 : MATRIX ALGEBRA 64 Properties of Transposes (1) (AT )T = A (2) (A + B )T = AT + B T when A + B exists (3) (λ A)T = λ AT for any λ ∈ R (4) (AB )T = B T AT when AB exists. Note 8 : MATRIX ALGEBRA Inverse Matrix • Only square matrices have inverses. 65 Note 8 : MATRIX ALGEBRA Inverse Matrix • Only square matrices have inverses. • Matrix B is the inverse of matrix A if A and B are square matrices of the same order and if AB = I = BA. 66 Note 8 : MATRIX ALGEBRA Inverse Matrix • Only square matrices have inverses. • Matrix B is the inverse of matrix A if A and B are square matrices of the same order and if AB = I = BA. • If A has an inverse, then that inverse is unique. 67 Note 8 : MATRIX ALGEBRA Inverse Matrix • Only square matrices have inverses. • Matrix B is the inverse of matrix A if A and B are square matrices of the same order and if AB = I = BA. • If A has an inverse, then that inverse is unique. • If A is the inverse of B then B is the inverse of A. 68 Note 8 : MATRIX ALGEBRA Inverse Matrix • Only square matrices have inverses. • Matrix B is the inverse of matrix A if A and B are square matrices of the same order and if AB = I = BA. • If A has an inverse, then that inverse is unique. • If A is the inverse of B then B is the inverse of A. • The inverse of matrix A is denoted by A−1 . 69 Note 8 : MATRIX ALGEBRA Inverse Matrix • Only square matrices have inverses. • Matrix B is the inverse of matrix A if A and B are square matrices of the same order and if AB = I = BA. • If A has an inverse, then that inverse is unique. • If A is the inverse of B then B is the inverse of A. • The inverse of matrix A is denoted by A−1 . More about how to calculate A−1 in a minute! 70 Note 8 : MATRIX ALGEBRA 71 The Determinant of a 2x2 Matrix • The determinant of a 2 × 2 matrix A = a b cd is defined to be the number ad − bc. a b . • The determinant is denoted by det(A), | A | or c d Note 8 : MATRIX ALGEBRA Examples " (1) If A = 2 −3 1 2 # then det(A) = (2 × 2) − ((−3) × 1) = 4 + 3 = 7. 72 Note 8 : MATRIX ALGEBRA Examples " # (1) If A = 2 −3 then det(A) = (2 × 2) − ((−3) × 1) = 4 + 3 = 7. 1 2 1 0 (2) = (1 × −1) − (0 × 0) = −1 + 0 = −1. 0 −1 73 Note 8 : MATRIX ALGEBRA Examples " # (1) If A = 2 −3 then det(A) = (2 × 2) − ((−3) × 1) = 4 + 3 = 7. 1 2 1 0 (2) = (1 × −1) − (0 × 0) = −1 + 0 = −1. 0 −1 " # " # " # (3) If A = a b and B = x y then AB = ax + bz ay + bt , so cd z t cx + dz cy + dt det(AB ) = (ax + bz )(cy + dt) − (ay + bt)(cx + dz ) = axcy + axdt + bzcy + bzdt − aycx − aydz − btcx − btdz = ad (xt − yz ) − bc(xt − yz ) = (ad − bc)(xt − yz ) = det(A) det(B ) 74 Note 8 : MATRIX ALGEBRA Examples " # (1) If A = 2 −3 then det(A) = (2 × 2) − ((−3) × 1) = 4 + 3 = 7. 1 2 1 0 (2) = (1 × −1) − (0 × 0) = −1 + 0 = −1. 0 −1 " # " # " # (3) If A = a b and B = x y then AB = ax + bz ay + bt , so cd z t cx + dz cy + dt det(AB ) = (ax + bz )(cy + dt) − (ay + bt)(cx + dz ) = axcy + axdt + bzcy + bzdt − aycx − aydz − btcx − btdz = ad (xt − yz ) − bc(xt − yz ) = (ad − bc)(xt − yz ) = det(A) det(B ) (4) Similarly, det(A1 A2 . . . An ) = det(A1 ) det(A2 ) . . . det(An ) for any matrices A1 , A2 , . . . An s.t. A1 A2 . . . An exists. 75 Note 8 : MATRIX ALGEBRA Question: Do all square matrices have inverses? 76 Note 8 : MATRIX ALGEBRA Question: Do all square matrices have inverses? Answer: No! 77 Note 8 : MATRIX ALGEBRA Question: Do all square matrices have inverses? Answer: No! Theorem Every matrix is invertible iff its determinant is nonzero. 78 Note 8 : MATRIX ALGEBRA Question: Do all square matrices have inverses? Answer: No! Theorem Every matrix is invertible iff its determinant is nonzero. Proof: • Suppose first that matrix A is invertible. 79 Note 8 : MATRIX ALGEBRA Question: Do all square matrices have inverses? Answer: No! Theorem Every matrix is invertible iff its determinant is nonzero. Proof: • Suppose first that matrix A is invertible. – We know that det(A) det(A−1 ) = det(AA−1 ) = det(I ) = 1. 80 Note 8 : MATRIX ALGEBRA Question: Do all square matrices have inverses? Answer: No! Theorem Every matrix is invertible iff its determinant is nonzero. Proof: • Suppose first that matrix A is invertible. – We know that det(A) det(A−1 ) = det(AA−1 ) = det(I ) = 1. – This means that det(A) 6= 0. 81 Note 8 : MATRIX ALGEBRA Question: Do all square matrices have inverses? Answer: No! Theorem Every matrix is invertible iff its determinant is nonzero. Proof: • Suppose first that matrix A is invertible. – We know that det(A) det(A−1 ) = det(AA−1 ) = det(I ) = 1. – This means that det(A) 6= 0. • Suppose now that matrix A had a nonzero determinant. 82 Note 8 : MATRIX ALGEBRA Question: Do all square matrices have inverses? Answer: No! Theorem Every matrix is invertible iff its determinant is nonzero. Proof: • Suppose first that matrix A is invertible. – We know that det(A) det(A−1 ) = det(AA−1 ) = det(I ) = 1. – This means that det(A) 6= 0. • Suppose now that matrix A had a nonzero determinant. – We will discuss this result in general in Note 10. 83 Note 8 : MATRIX ALGEBRA The Inverse of a 2x2 Matrix A 2 × 2 matrix is invertible iff its determinant is nonzero. " # " # If A = a b and det(A) 6= 0 then A−1 = det1 A d −b cd −c a 84 Note 8 : MATRIX ALGEBRA Examples Calculate, where possible, the inverses of the following 2x2 matrices: " # (1) A = 1 1 02 " # (1) B = 1 4 12 " # (1) C = 3 −1 1 3 " # (1) D = −1 4 2 −8 85
Related documents