Inverse (reverse) matrices have basic properties, but before that, we first need to get to know several types of matrices that will be used directly.
A matrix is said to be a zero matrix if all elements of the matrix are zero, while the size of the zero matrix depends on its partner matrices.
Getting to Know Matrix Inverse
EXAMPLE 1:
Examples of several zero matrices with several different sizes
If an arbitrary matrix A and a zero matrix 0 are of the same size, it is clear that A + 0 = 0 + A = A, just as the real numbers a + 0 = 0 + a = a. Three numbers a, b and c are all nonzero, if ab = ac, then b = c, and likewise for two different numbers, if de =, then one of them must be zero. This is not true for matrices.
EXAMPLE 2:
applicable
Even though matrix B is not the same as matrix C, and AD = 0, one of the matrices does not have to be zero.
The identity matrix is a square matrix whose elements are all zero except those on the main diagonal which are all ones, usually symbolized by In, where n is the size of the matrix.
EXAMPLE 3:
Some examples of identity matrices
If an arbitrary matrix A is multiplied by an identity matrix or vice versa (which can be done), the result is a matrix A itself, or written AI = IA = A
EXAMPLE 4:
Suppose the matrix
so
so are
THEOREM 1 - If a square matrix A is OBE-ed to form a reduced echelon row matrix, namely R, then R is a matrix that has all zero rows or the identity matrix.
Proof:
Consider the square matrix A, then do OBE, every one of the main diagonals produced, then in that column, the rows are all zero. If done continuously, then what is produced is an identity matrix or a matrix containing all zero rows.
EXAMPLE 5:
perform elementary row operations, so that
In this section we will discuss the inverse of a matrix. First, pay attention to the definition of inverse below.
THEOREM 2 - If A is a square matrix and if another square matrix B can be found of the same size, such that AB = BA = I, then A is called the inverse matrix or the matrix that has an inverse and the matrix B is called the inverse of the matrix A.
EXAMPLE 6:
Because
And
Now consider the following theorem
THEOREM 3 - If B and C are both inverses of matrix A, then B = C
Proof:
Since B is the inverse of A, then AB = I. Multiply both sides by C, so that C(AB) = CI = C, while (CA)B = IB = B, so B = C
THEOREM 4 - If matrices A and B are matrices that have inverses and are the same size, then;
- AB also has an inverse
(AB)*-1* = B*-1*A*-1*
Proof:
By multiplying both sides by AB, then
EXAMPLE 7:
then it can be found
whereas
Some of the properties implied in the definitions and theorems (look for proof in other books) that can be used to increase insight include:
THEOREM 5 - If a square matrix A, then it can be defined
if A has an inverse, it is defined
THEOREM 6 - If a square matrix A, and r; s are integers, then
THEOREM 7 - If A is a matrix that has an inverse, then
EXAMPLE 8:
adopting from the matrix in Example 1.4.7, namely
so
And
THEOREM 8 - If matrix A has an inverse, then AT also has an inverse and
EXAMPLE 9:
adopting from the matrix in Example 1.4.7, namely
so
as in Theorem 8.
Matrix & Its Operations
Matrix Definition & Applicable Operations - Matrices have several properties that apply to their elements, namely they are limited to real numbers only. Take a look at some of the definitions below!
DEFINITION 1 - A matrix is a rectangular arrangement of numbers. The numbers in the arrangement are called elements of the matrix.
Getting to Know Matrices and Their Operations
EXAMPLE 1;
The size of the matrix is indicated by the number of rows and columns, as above in sequence, the size of the first matrix is 3 x 2, because the matrix consists of three rows and two columns. Likewise, the next matrix has a size of 3 x 3, the third matrix is also called a row matrix or row vector because it only consists of one row and the last is a column matrix or column vector, because it only consists of one column. Both, column vectors and row vectors are symbolized by a bold lowercase letter or a lowercase letter with an underline.
In general, the notation for a matrix uses capital letters, while the members of the matrix usually use lowercase letters.
EXAMPLE 2;
Matrix A has size mxn, so the matrix can be written;
or it can also be written like this;
if you want to call a member of the matrix A in the i-th row and j-th column, namely
DEFINITION 2 - Two matrices are said to be equal if they have the same size and their corresponding members are also the same
If there are two matrices A = (aij) and B = (bij) which are said to be the same, then (A)ij = (B)ij applies. Consider the example below.
EXAMPLE 3;
If matrix A = B, then the value of x in A must be equal to 2. Matrix B is not the same as matrix C, because the two matrices do not have the same size.
DEFINITION 3 - If two matrices A and B have the same size, then they can be added or subtracted. To add or subtract the two matrices, the corresponding members are added or subtracted. Matrices that do not have the same size cannot be added or subtracted.
Two matrices A = (aij) and B = (bij) can be added or subtracted if the two matrices have the same size, the result of the addition or subtraction is
EXAMPLE 4;
then the result of adding and subtracting matrices A and B is
what if A + C, it cannot be done because the size of the two matrices is not the same.
DEFINITION 4 - If A is any matrix and c is any scalar, then the product of the scalar and the matrix cA is to multiply all elements of A by the scalar c
EXAMPLE 5;
If the matrix A in EXAMPLE 4 is multiplied by 3, the result is
Likewise, if the matrix C is multiplied by 2, the result is
DEFINITION 5 - Two matrices A and B can be multiplied, if matrix A has rxn, and matrix B must have size nxl then the resulting matrix has size rxl with the ijth element coming from the multiplication of the i-th row of matrix A with the j-th column of matrix B.
EXAMPLE 6;
Matrices A and B in Example 1.3.4 can be multiplied, because the size of matrix A is 2 £ 2 and matrix B is 2 £ 2 so the two matrices can be multiplied and the result is
in the same way, if matrix A is multiplied by matrix C, the result is
while matrix C cannot be multiplied by matrix A, because the size of the matrix does not match the existing definition.
DEFINITION 6 - The transpose matrix of matrix A is written AT whose members are the members of A by changing rows into columns and columns into rows.
EXAMPLE 7;
The transpose of the three matrices in EXAMPLE 4 is
DEFINITION 7 - If matrix A is square, then the trace of A is given by tr(A), defined as the sum of the elements on the main diagonal of matrix A.
EXAMPLE 8;
By using the matrix in EXAMPLE 4, then
while the trace of matrix C cannot be searched, because matrix C is not a square matrix.
Matrix Theory Problem Solving Example
If A is any 4x4 matrix, if A' is a matrix that is produced when a single row of A is multiplied by const. K, then it applies >> det(A') = k.det(A)
- If A is the matrix that results when 2 rows of A are exchanged then det|A| applies = -det(A) or det(A') = -det(A)
- If A is the matrix that results when a multiple of a row A is added to another row then det(A') = det(A) applies.
Calculate det A, by reducing the matrix A to row escon form using Theorem 2.
Completion
Summarized on October 26, 2014