chapter 7: matrices

Chapter 7 Matrices Chapter 7 Matrices Nguyen Thi Minh Tam ntmtam vnuagmail com December 10, 2020 1 7 1 Basic matrix operations 2 7 2 Matrix Inversion 3 7 3 Cramer’s rule 7 1 Basic matrix operations Matrix A matrix of order m ×n is rectangular array of numbers with m rows and n columns The numbers constituting the array are called entries or elements If A is an m ×n matrix, and if the element in row i and column j of A is denoted by aij , then A is written as follow A =   a11 a12 a1n a21 a2.

Chapter 7: Matrices

Nguyen Thi Minh Tamntmtam.vnua@gmail.comDecember 10, 2020

1 7.1 Basic matrix operations

2 7.2 Matrix Inversion

3 7.3 Cramer’s rule

7.1 Basic matrix operations

Matrix

A matrix of order m × nis rectangular array of numbers with

m rows and n columns

The numbers constituting the array are called entriesor

. .am1 am2 amn

Example 1 Let A =1 2

3 4

, B =



a) State the orders of the matrices A and B

b) Write down the values of a11, a22, b25, b33, b34

Equality of Matrices

Two matrices A and B are calledequal( written A = B ) if

(1) they have the same order,

(2) corresponding entries are equal

Square Matrix, Zero Matrix, Negative Matrix

A matrix of order n × n is called a square matrixof order n.The m × n matrix in which every entry is zero is called the

zero matrix, denoted as 0

If A = [aij]m×n, then the negative matrixof A ( written –A) isdefined by

−A = [−aij]m×n

Row vector, column vector, zero matrix

A matrix that has only one row is called arow vector

A matrix that has only one column is called acolumn vector

Transpose of a Matrix

If A is an m × n matrix, thetransposeof A, written AT, is the

n × m matrix whose rows are just the columns of A in the sameorder

Example 4 Write down the transpose of the matrix

A =1 2 3

4 5 6



Matrix Addition and Subtraction

If A = [aij]m×n, B = [bij]m×n, then

thesum of A and B, denoted A + B, is defined by

A + B = [aij+ bij]m×nthedifference of A and B, denoted A − B, is defined by

A − B = [aij− bij]m×n

Note To add (or subtract) two matrices of the same order, wesimply add (or subtract) their corresponding elements

Example 5 (1 page 540)

The monthly sales (in thousands) of burgers (B1) and bites (B2)

in three fast-food restaurants (R1, R2, R3) are as follows:

a) Write down two 2 × 3 matrices J and F , representing sales inJanuary and February, respectively

b) By finding J + F , write down the matrix for the total salesover the two months.

c) By finding J–F , write down the matrix for the difference insales for the two months

Properties of matrix addition and subtraction

If A, B, C are any matrices of the same order, then

(1) A + B = B + A (commutative law)

(2) A + 0 = A

(3) A − A = 0

Properties of Scalar Multiplication

If A, B are any matrices of the same order and k, p are any realnumbers, then

(1) k(A + B) = kA + kB

(2) (k + p)A = kA + pA

(3) (kp)A = k(pA)

Matrix multiplication

If R is a row vector

R =r1 r2 rnand C is the column vector

If A is an m × s matrix and B is an s × n matrix, then





112





Find (where possible) ac, bd , ad

The product AB exists if the number of columns of A equalsthe number of rows of B

In general, AB 6= BA

Properties of Matrix Multiplication

Assume that k is an arbitrary scalar and A, B, C are matrices oforders such that the indicated operations can be performed Then

1) A(BC ) = (AB)C (associative law)

2) A(B + C ) = AB + AC , A(B–C ) = AB–AC (distributive law)

3) (B + C )A = BA + CA, (B–C )A = BA–CA (distributive law)

4) k(AB) = (kA)B = A(kB)

5) (AB)T= BTAT

Matrix form of a system of linear equations

Asystem of m linear equations in n variablesis a system of theform

a11x1+ a12x2+ + a1nxn= b1a21x1+ a22x2+ + a2nxn= b2

am1x1+ am2x2+ + amnxn= bm

.xn

.bm

the system (1) can be written inmatrix form

AX = bA: thecoefficient matrix, b: the right-hand-side vector

Example 9 The system of equations

2x + 3y − 2z = 6

x − y + 2z = 34x + 2y + 5z = 1can be expressed in the form AX = b Write down the matrices A,

X and b

Example 10.

I2 =1 0

0 1

, I3=

Thedeterminantof the matrix A =a b

c d , denoted by det(A) or

|A|, is defined as follows

det(A) =

a b

c d

Inverses and Linear Systems

Suppose a system of n linear equations in n variables is written inmatrix form as

AX = b

If A is invertible, the system has the unique solution

X = A−1b

Example 12 For a closed economy with no government

intervention, the equilibrium levels of consumption C and income

Y satisfy the system

Y = C + I∗

C = aY + bwhere a and b are parameters (0 < a < 1 and b > 0), and I∗denotes investment

a) Express this system in the matrix form, where the variablesare Y , C

b) Find the value of Y , C in terms of a, b, I∗

Consider the consumption function

1 − a∆b

Therefore, theautonomous consumption multiplier for Y is 1

1 − aand theautonomous consumption multiplier for C is also 1

1 − a.

Example 13 Find all the cofactors of the matrix

Assume that determinants of (n − 1) × (n − 1) matrices have beendefined Given the n × n matrix A = [aij] The determinant of A isdefined by

det(A) =a11A11+a12A12+ +a1nA1n

This is called thecofactor expansionof det A along row 1

Note det(A) can be computed by expanding along any row orcolumn: multiply each entry of that row or column by the

corresponding cofactor and add

Example 14 Find the determinant of the matrices

The inverse of a square matrix of order 3



T

Example 15 Find the inverses of

P1+ 3P2+ 4P3 = 35

det(An)det(A)where, for each i , Ai is the matrix obtained from A by replacingcolumn i by b.

Example 17 Consider the following macroeconomic model:

Y = C + I∗+ G∗+ X∗− M∗ (quilibrium of national income)



 and

A and b are 3 × 3 and 3 × 1 matrices to be stated

b) Use Cramer’s rule to solve this system for Y

c) Find the autonomous investment multiplier for Y and deducethat Y increases as I∗ increases

Exercise 7.1, 7.1* 4, 5, 6 (page 540, 541) 4, 5 (page 543)

Exercise 7.2, 7.2* 6(page 561), 5, 6, 7 (page 562)

Exercise 7.3, 7.3* 5, 6, 7 (page 573), 4, 5, 6, 7 (page 574-575)

Assume that determinants of (n − 1) × (n − 1) matrices have beendefined Given the n × n matrix A = [aij] The determinant of A isdefined... class="text_page_counter">Trang 30

Example 14 Find the determinant of the matrices< /p>Trang 31

The... M∗ (quilibrium of national income)



 and

A and b are × and × matrices to be stated

b) Use Cramer’s rule to solve this system for Y

c) Find the