THE headquarters it is commonly used for organizing tabular data to facilitate problem solving. Matrix information, whether numeric or not, is arranged neatly in rows and columns.
The set of matrices equipped with the operations of addition, subtraction and multiplication and features, as a neutral and inverse element, form a mathematical structure that enables its application in various fields of this large area of knowledge.
See too: Relationship between matrix and linear systems
Matrix representation
Before starting the studies on matrices, it is necessary to establish some notations regarding their representations. At matrices are always represented by capital letters. (A, B, C…), which are accompanied by indexes, in which the first number indicates the number of rows, and the second, the number of columns.
THE number of lines (horizontal rows) and columns (vertical rows) of a matrix determines its order. The matrix A has order m by n. The information contained in an array is called elements and are organized in parentheses, square brackets or two vertical bars, see the examples:
The matrix A has two rows and three columns, so its order is two by three → A2x3.
Matrix B has one row and four columns, so its order is one by four, so it is called line matrix → B1x4.
Matrix C has three rows and one column, and so it is called column matrix and its order is three by one → C3x1.
We can generically represent the elements of an array, that is, we can write this element using a mathematical representation. Ogeneric element will be represented by lowercase letters (a, b, c…), and, as in the representation of arrays, it also has an index that indicates its location. The first number indicates the row the element is in, and the second number indicates the column in which it is located.
Consider the following matrix A, we will list its elements.
Observing the first element that is located in the first row and first column, that is, in row one and column one, we have the number 4. In order to make writing easier, we will denote it by:
The11 → line one element, column one
So we have the following elements of matrix A2x3:
The11 = 4
The12 =16
The13 = 25
The21 = 81
The22 = 100
The23 = 9
In general, we can write an array as a function of its generic elements, this is the generic matrix.
A matrix of m row and n columns is represented by:
Example
Determine the matrix A = [aij ]2x2, which has the following training law toij = j2 – 2i. From the statement data, we have that the matrix A is of order two by two, that is, it has two lines and two columns, therefore:
In addition, the matrix formation law was given, that is, each element is satisfied with the relation toij = j2 – 2i. Substituting the values of i and j in the formula, we have:
The11 = (1)2 - 2(1) = -1
The12 = (2)2 - 2(1) = 2
The21 = (1)2 - 2(2) = -3
The22 = (2)2 - 2(2) = 0
Therefore, matrix A is:
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Array Types
Some matrices deserve special attention, see now these types of arrays with examples.
square matrix
A matrix is square when the number of rows equals the number of columns. We represent the matrix that has n rows and n columns by Ano (reads: square matrix of order n).
In square matrices, we have two very important elements, the diagonals: main and secondary. The main diagonal is formed by elements that have equal indices, that is, it is every element aij with i = j. The secondary diagonal is formed by elements aij with i + j = n +1, where n is matrix order.
identity matrix
The identity matrix is a square matrix that has allyouelements of the main diagonal equal to 1 and the other elements equal to 0, its formation law is:
We denote this matrix by I, where n is the order of the square matrix, see some examples:
unit matrix
It is a square matrix of order one, that is, it has a row and a column and, therefore, just one element.
A = [-1]1x1, B = I1 = (1)1x1 and C = || 5||1x1
These are examples of unitary matrices, with emphasis on matrix B, which is a unit identity matrix.
null matrix
An array is said to be null if all its elements are equal to zero. We represent a null matrix of order m by n by Omxn.
The matrix O is null of order 4.
opposite matrix
Consider two equal-order matrices: A = [aij]mxn and B = [bij]mxn. These matrices will be called opposite if, and only if, theij = -bij. Thus, the corresponding elements must be opposite numbers.
We can represent the matrix B = -A.
transposed matrix
Two matrices A = [aij]mxn and B = [bij]nxm they are transposed if, and only if, theij = bji , that is, given a matrix A, to find its transpose, just take the lines as columns.
The transpose of matrix A is denoted by AT. See the example:
See more: Inverse matrix: what is it and how to verify
Matrix operations
The set of matrices has the operations of avery well defined addition and multiplication, that is, whenever we operate two or more matrices, the result of the operation still belongs to the set of matrices. However, what about the subtraction operation? We understand this operation as being the inverse of addition (opposite matrix), which is also very well defined.
Before we define the operations, let's understand the ideas of corresponding element and equality of matrices. Corresponding elements are those that occupy the same position in different matrices, that is, they are located in the same row and column. Obviously the arrays need to be of the same order for matching elements to exist. Look:
Elements 14 and -14 are corresponding elements of opposite matrices A and B, as they occupy the same position (same row and column).
Two matrices will be said to be equal if and only if the corresponding elements are equal. Thus, given the matrices A = [aij]mxn and B = [bij]mxn, these will be the same if, and only if, theij = bij for any i j.
Example
Knowing that matrices A and B are equal, determine the values of x and t.
Since matrices A and B are equal, then the corresponding elements must be equal, therefore:
x = -1 and t = 1
Addition and subtraction of matrices
The operations of addition and subtraction between matrices they are quite intuitive, but first a condition must be satisfied. To perform these operations, it is first necessary to verify that the array orders are equal.
Once this condition is verified, the addition and subtraction of the matrix takes place by adding or subtracting the corresponding elements of the matrices. Consider the matrices A = [aij]mxn and B = [bij]mxn, then:
A + B = [aij + bij] mxn
A - B = [aij - Bij] mxn
Example
Consider matrices A and B below, determine A + B and A – B.
Read too: Whole number operations
Multiplication of a real number by matrix
The multiplication of a real number in a matrix (also known as matrix multiplication) by a scalar is given by multiplying each element of the matrix by the scalar.
Let A = [aij]mxn a matrix and t a real number, so:
t · A = [t · aij]mxn
See the example:
Matrix multiplication
The multiplication of matrices is not as trivial as the addition and subtraction of them. Before performing the multiplication, a condition must also be satisfied regarding the order of the matrices. Consider matrices Amxn and Bnxr.
To perform the multiplication, the number of columns in the first matrix must equal the number of rows in the second. The product matrix (which comes from multiplication) has an order given by the number of rows in the first and the number of columns in the second.
To perform the multiplication between matrices A and B, we must multiply each of the rows by all the columns as follows: the first element of A is multiplied by the first element of B and then added to the second element of A and multiplied by the second element of B, and so successively. See the example:
Read too: Laplace's Theorem: know how and when to use
solved exercises
question 1 – (U. AND. Londrina – PR) Let the matrices A and B be, respectively, 3 x 4 and p x q, and if the matrix A · B has order 3 x 5, then it is true that:
a) p = 5 and q = 5
b) p = 4 and q = 5
c) p = 3 and q = 5
d) p = 3 and q = 4
e) p = 3 and q = 3
Solution
We have the statement that:
THE3x4 · Bpxq = C3x5
From the condition to multiply two matrices, we have that the product only exists if the number of columns in the first is equal to the number of rows in the second, so p = 4. And we also know that the product matrix is given by the number of rows in the first with the number of columns in the second, so q = 5.
Therefore, p = 4 and q = 5.
A: Alternative b
Question 2 - (Vunesp) Determine the values of x, y, and z, on the following equality, involving 2 x 2 real matrices.
Solution
Let's perform the operations between the arrays and then the equality between them.
To determine the value of x, y and z, we will solve the linear system. Initially, let's add equations (1) and (2).
2x – 4= 0
2x = 4
x = 2
Substituting the value of x found in equation (3), we have:
22 = 2z
2z = 4
z = 2
And finally, substituting the values of x and z found in equation (1) or (2), we have:
x + y - z = 0
2 +y – 2 = 0
y=0
Therefore, the solution to the problem is given by S = {(2, 0, 2)}.
by Robson Luiz
Maths teacher
