Chapter 1 matrix system determinant (1)

Matrix, System of linear equations, Determinant Phan Thi Khanh Van E-mail: khanhvanphan@hcmut.edu.vn May 12, 2021 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 / 66 Table of contents Matrices Elementary row (column) operations of a matrix Rank of matrices System of linear equations The homogeneous system of linear equations Operations with matrices Inverse of a matrix Determinant (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 / 66 Example (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 / 66 Example Consider a system of linear equations     −1  x + 3y − z = 3x − 4y + 2z = −3 →  −4 −3    2 2x + y =2 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 / 66 Matrix A matrix A of order m × n (A is m by n) is a rectangular array of numbers and/or variables with m rows and n columns   a11 a12 a1j a1n  a21 a22 a2j a2n        A=   ai1 ai2 aij ain      am1 am2 amj amn Denote A = (aij )m×n The set of all m × n matrices over the field K (R, C) is denoted by Mm×n [K] (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 / 66 Examples −2 −1  −i + −3 2 × matrix  1 - × matrix i 1 -1 × matrix - row matrix (row vector)   −3 - × matrix - column matrix (column vector) (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 / 66 Zero matrix is the matrix whose all entries are 0: (aij = 0), ∀1 ≤ i ≤ m, ≤ j ≤ n Denote 0m×n 02×3 = 0 0 0 Transpose of a matrix The transpose AT of a matrix A is an operator which switches the row and column indices of the matrix: [AT ]ij = aji If A is m × n, then AT is n × m   −3 A= ⇒ AT = 1  −3 1 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 / 66 Square matrix A square matrix is a matrix with the same number of rows and columns m = n An n-by-n matrix is known as a square matrix of order n The set of all square matrices of order n over the field K is denoted by Mn [K] Example:  A = −3   - a square matrix of order The entries aii form the main diagonal of a square matrix The sum of the main diagonal a11 + a22 + + ann is called the trace, denoted by tr (A), trace(A) (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 / 66 Upper triangular matrix A square matrix is called an upper triangular matrix if all the entries below themain diagonal are zeros  Example A = 0 1 0 Lower triangular matrix A square matrix is called an lower triangular matrix if all the entries above  diagonal are zeros  the main 0  Example: A = −1 0 −2 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 / 66 Diagonal matrix A matrix that is both upper and lower triangular is called a diagonal matrix:  aij = 0,∀i = j 0  Example: A = −1 0 0 Identity matrix (unit matrix) The identity matrix of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere: aij = 0, ∀i = j, a ii = 1, ∀i 0 Example: I3 = 0 0 0 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 10 / 66 Application of matrix in cryptography Chose a square matrix of order 3: K as the ”symmetric key” and thenmultiply it to the left of A:   1 1 18 27 K = 2 0 ⇒ B = K A = 2 0 27 26 0 0 21 18 25   39 45 26  = 63 57 28 21 18 25 We have the encrypted series of numbers 39 63 21 45 57 18 26 28 25 We can recover A from B by: A = K −1 B (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 52 / 66 Input output Leontief model Example Consider a very simple economy that runs on different types of output: raw materials, services, and manufacturing Each industry requires some amount of output from each of the three to its job All of these requirements can be summarized in the following table: Industry Raw materials Services Manufacturing Raw materials 0.02 0.04 0.04 Services 0.05 0.03 0.01 Manufacturing 0.2 0.01 0.1 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 53 / 66 The numbers in the table tell how much output from each industry a given industry requires in order to produce one dollar of its own output For example, to provide 1$ worth of service, the service sector requires 0.04$ worth of raw materials, 0.03$ worth of services, and 0.01$ worth of manufactured goods The demand matrix D tells how much ( in billions of dollars) of each type of output is demanded and others  outside the economy   by consumers 400 0.02 0.04 0.04 D = 200, A = 0.05 0.03 0.01 Let 600 0.2 0.01 0.1 T X = (x, y , z) denote the production matrix It represents the amounts (in billions of dollars of value) produced by each of the three industries (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 54 / 66 We have the equation: Internal demand + External demand = Total production For ex.: + The amount of money needed in Raw materials industry to produce $x of Raw materials, $y of Services and $z of Manufacturing: 0.02x + 0.04y + 0.04z + The external demand of Raw materials industry: 400 ⇒ 0.02x + 0.04y + 0.04z + 400 = x Hence, we have AX + D = X Therefore, X = (I − A)−1 D =  −1     0.98 −0.04 −0.04 400 449.24 −0.05 0.97 −0.01 200 = 237.27 −0.2 −0.01 0.9 600 769.13 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 55 / 66 Determinant Every square matrix A can be associated with a number called its determinant Denote: |A| (or det(A)) k = : A = a11 ⇒ |A| = a11 a11 a12 a21 a22 ⇒ |A| = a11 a22 − a21 a12   a11 a12 a13 k = : A = a21 a22 a23  a31 a32 a33 a a ⇒ |A| = a11 (−1)(1+1) 22 23 + a32 a33 a a a a a12 (−1)(1+2) 21 23 + a13 (−1)(1+3) 21 22 a31 a32 a31 a33 k =2: A= (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 56 / 66 Example =5 3+i −2 − i 1 = (3 + i).4 − 1(−2 − i) = 14 + 5i -1 0 = 1.(−1)(1+1) + −2 −2 2 3.(−1)(1+2) + (−1).(−1)(1+3) −2 1 = −8 + 12 + = (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 57 / 66 The i, j cofactor of a matrix The i, j cofactor of A: Aij = (−1)i+j Mij where Mij - i, j minor of A (the determinant of the (n − 1) × (n − 1) matrix that results from deleting the i−th row and the j−th column of A) Determinant of an n × n matrix A is given by the expansion: A - n × n matrix Fixing any i, j ∈ {1, n} Then, |A| = ai1 Ai1 + ai2 Ai2 + ai3 Ai3 + + ain Ain |A| = a1j A1j + a2j A2j + a3j A3j + + anj Anj (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 58 / 66 Example 2 = 1.(−1)3+1 =0 −1 1 −2 −5 2+4 +1(−1) −5 (Phan Thi Khanh Van) + 3(−1)3+2 2+2 = 1.(−1) + 5(−1)3+3 4 −1 −2 −5 4 =0 Matrix, System of linear equations, Determinant May 12, 2021 59 / 66 Example −1 −1 −2 1+3 = (1) −2 = −11 0 0 −1 −1 2 1 −1 21 −4 −1 12 1 =0 11 15 64 −11 26 0 4 = = 2.4.(−2) = −16 −2 1 −2 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 60 / 66 Remark If A has at least proportional rows (columns), then |A| = If A has at least one zero row (column), then |A| = The determinant of a triangular matrix equals the product of diagonal entries Effects of Elementary Row Operations on Determinants r →αr i A −i−−→ B ⇒ |B| = α|A| A −−−−−→ B ⇒ |B| = |A| A −−−→ B ⇒ |B| = −|A| ri →ri +αrj ri ↔rj (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 61 / 66 Example: Using row operations to find the determinant of A −2 4 1 −2 4 |A| = =− −3 1 −1 −3 1 −1 −1 −5 −4 −1 −5 −4 r2 → r2 − 2r1 −2 −7 r3 → r3 + 3r1 −7 − = − −5 11 r4 → r4 + r1 −5 11 −7 −4 = −7 −4 −7 r2 → r2 + r1 = − 51 = 127 r3 → 5r3 + 7r1 −19 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 62 / 66 Properties of the determinant |In | = |AT | = |A| |AB| = |A|.|B| |An | = |A|n |αA| = αn |A| In general, |A + B| = |A| + |B| |A−1 | = |A| A is invertible ⇔ rank(A) = n ⇔ |A| = (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 63 / 66 Example Let A and B be × matrices such that |A| = 3, |B| = Find |2A2019 B −1 | |2A2019 B −1 | = 23 |A2019 ||B −1 | (Because A.B are × = 23 32019 12 = 4.32019 matrices) = 23 |A|2019 |B| (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 64 / 66 Example Find m such that the following system of linear equations  x + 2y + z = has unique solution 2x − 3y + mz = (∗)   4x − 2y + z = (∗) is a square system: equations with variable, it has unique solution ⇔ rank(A) = rank(A|b) = ⇔ |A| = 1 ⇔ −3 m = ⇔ 10m + = ⇔ m = − 10 −2 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 65 / 66 Thank you for your attention! (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12, 2021 66 / 66 ... -4 −7  ? ?1 −7 12 39 −33 ? ?15  0 −4 −7  12 0 −45 39 21 12 39 −33 ? ?15  0 −4 −7   ? ?15 13 13 0 ? ?15 4 4 ? ?11 −5  ? ?1  13 − 11 ⇒ A = 13 4 4 −3 −5 0 74 −5 4 4 (Phan Thi Khanh Van) Matrix, System of... = 10 0  ? ?1 0 0 1 ↔  1 ? ?1 0 0 1 (Phan Thi Khanh Van)  ? ?1 0 0 1 ↔  1 ? ?1 0 0 1  400 300   200  10 0 Matrix, System of linear equations, Determinant  400 300   600  10 0 May 12 , 20 21. .. equations, Determinant May 12 , 20 21 41 / 66 Example Given A = 1 A3 = An = A2 = 1 n 1 Find An 1 1 = 1 1 = 1 (Phan Thi Khanh Van) Matrix, System of linear equations, Determinant May 12 , 20 21 42 /