Matrix Equations and Inverse

Question 1

what is a matrix?

Answer 1

is a double array of numbers written in the rectangular form

ex) A = [1 2 3]
- 1 2 -2 1 0

A has 2 rows and 3 columns

dimensions of a matrix: the number of rows and the number of columns are called the dimensions of the matrix

If a has m rows and n columns, we say A is an m x n matrix

we denote matrices by upper English letters A, B, C

to denote entries of a matrix we use lower case English letter of the name of the matrix

Question 2

matrix operations

Answer 2

quality of matrices: two matrices A and B are equal A=B if

1) they have same dimensions
2) the corresponding entries are the same

A = [a_ij]_mxn B = [b_ij]_kxl

A = B <=> { m = k and n = l

a_ij = b_ij for all i and j

sum of matrices: we can add any two matrices with the same dimensions

multiplication by a scalar: let C be a number and A be a m x n matrix A = [a_ij]_nxm

negative of a matrix: -A = (-1)A = [-a_ij]

matrix subtraction: A_nxm - B_nxm = A + (-B) = A + (-1)B = [a_ij - b_ij]

Question 3

transpose matrix

Answer 3

definition: let A = [a_ij] be an mxn matrix. transpose of A, A^T is a matrix

1) with n rows and m columns (A^T)_nxm = (A_nxm)^T
2) (i, j) entry of A^T is (j, i) entry of A

properties:

1) (A^T)^T = A
2) (A + B)^T = A^T + B^T
3) (cA)^T = cA^T

Question 4

matrix multiplication

Answer 4

1) dimension compatibility: two matrices can be multiplied if # columns of the first matrix = # row of the second matrix
2) dimesions of the product matrix:

rows of product = # rows of the first matrix

columns of product = # columns of the second matrix

3) entries of the product matrix

(i, j) - entry of the product = (i^th row of the first matrix j^th) * (column of second)

remark 1) if the compatibility fails the matrix, multiplication is not defined

remark 2) compatibility condition makes the components of rows of the first matrix, equal to number of components of columns of the second matrix

Question 5

what is the inverse of a matrix

Answer 5

Q1: what is the inverse of a number a? the inverse of a is a number b such that

ab = 1 2 (1/2) = 1 (-1)(-1) = 1

Q2: does ever number have an inverse? every nonzero number has an inverse

a = 0 ab = 0b = 0 doesnt = 1

Q3: how do we find the inverse of a number? 1/a

a doesnt = 0 a(1/a) = a/a = 1

definition: let A be a square matrix of order n (nxn) if there exists a square matrix B such that

AB = I = BA where I is the identity matrix of order n, then we say A is invertible and B is called the inverse of A, denoted by A^-1

* the inverse of an invertible matrix is unique

the zero square matrix is not invertible

[1 0]

0 0 is not invertible

Question 6

finding the inverse of a matrix: using row reduction

Answer 6

Let A be a square matrix of order n

1) form an augmented matrix [A | I]_nx2n

2) reduce it to RREF [C | D]

3) there are two cases - C = I then A is invertible A^-1 = D [A | I] => [I | A^-1]

• C is not equal to I then A is not invertible

* if RREF of A is I then matrix A is invertible

Question 7

matrix form of SLE

Answer 7

Question 8

inverse matrix method

Answer 8

recall: matix form of SLE

in the method of inverses

1) system in its matrix form Ax = b

2) find the inverse of A, A^-1

3) the SLE has unique solution given by x = A^-1b

Question 9

matrix powers

Answer 9

when can a matrix be multiplied to itself?