Matrices have various operations such as addition, subtraction, and multiplication. These operations use their own way. This is caused by the arrangement of numbers in the matrix which is different from other mathematical operations.
The following is an explanation of the addition operation on the matrix and examples of questions to support the You learning process.
Overview of the Matrix
A matrix is an arrangement of numbers, symbols or expressions arranged in columns and rows to form a rectangular shape. As an initial description of the matrix, You can listen to the example of a 2 x 3 matrix below.
The size of the matrix is determined by the number of rows and columns it has. A matrix with m columns and n rows is called an m x n matrix, where m and n are called by their dimensions. For example, the matrix above is called a 2 x 3 matrix. This is because the matrix consists of 2 rows and 3 columns.
A matrix with the same number of rows and columns is called a square matrix. The matrix with the number of one row is called a row vector. Meanwhile, a matrix with one column is called a column vector.
An infinite matrix is a matrix with an unlimited number of rows or columns (or both). In some contexts, matrices considered without rows or columns are called empty matrices.
For further explanation, You can listen to the image below.
Row m is horizontal and column n is vertical. Each element of the matrix is often denoted by a variable of two index notations. For example, a 2,1 represents the elements in the second row and first column of matrix A.
Each object in the matrix A with dimension m x n is often denoted by a i,j. Which is the maximum value i = m and the maximum value j = n. The objects in the matrix are called elements, entries or members of the matrix.
If two matrices have the same dimensions (each matrix has the same number of rows and columns), then the two matrices can be added or subtracted element by element. However, based on the rules of matrix multiplication, the conditions for matrix multiplication, namely when the number of columns of the first matrix is equal to the number of rows of the second matrix in the multiplication of two matrices.
That is, multiplying an m x n matrix by an n x p matrix results in an m x p matrix . Therefore, matrix multiplication is not commutative. In general, matrices are used to represent linear transformations, that is, a generalization of linear functions such as f ( x ) = 4 x.
For example, the effect of rotation in three-dimensional space is a linear transformation denoted by the matrix R. If v is a vector in three dimensions, the result R v represents the position of the point after it is rotated.
The matrix can be applied in various cyan fields. For example in physics in the form of classical mechanics, optics, and quantum mechanics. Matrices are also used to study physical states, such as the motion of planets. In the field of computer graphics, matrices are applied to manipulate 3D models and project them onto a two-dimensional screen.
In the field of probability theory and statistics, matrices are used as an explanation of state probabilities. As in the pagerank algorithm in determining the order of disbursement pages on Google. The matrix calculus generalizes the classical analytic form of literal and exponential to higher dimensions. Matrices are also applied in economics to describe relational economic systems.
Matrix Functions in Everyday Life
Even though matrix operations look difficult, they have many benefits to make human work easier in everyday life. Here are some of the benefits of studying matrices in everyday life.
- Assist engineers in solving problems with many variables.
- Matrix can also be used to create reports and journals.
- Solving a system of linear equations, geometric transformations, determining television broadcast schedules, and computer programming.
- Helps analyze economic problems that have various kinds of variables.
- As a way to analyze in statistics, education, science, economics, and technology.
- Help find solutions to investigation operations, for example natural resource investigation operations (coal, petroleum, and so on).
Matrix Addition Concept
The addition operation on matrices can only be performed when the order of the matrices in the operation is the same. The sum of two matrices A = and B = is a new matrix C = of the same order, that is, its elements are the result of the addition or subtraction of the elements of matrices A and B.
The following is the concept or operating formula for matrix addition.
The properties of the matrix addition operation are as follows.
1. Commutative properties A + B = B = A 2. Associative nature (A + B) + C = A + (B + C) = A + B + C The null matrix is a summation identity matrix so it applies: A + 0 = 0 + A = A The identity matrix in the arithmetic operation of addition matrix –A. A + (-A) = (-A) + A = 0
In order to better understand matrix addition, You can listen to the following example questions.
Calculate A + B, if you know matrices A and B as follows.
Transpose, Determinant, and Matrix Inverse
1. Transpose the Matrix
Transpose matrix is a matrix that is operated by exchanging row elements into columns and column elements into rows from the initial matrix. The notation of the transpose matrix is usually with AT .
The transpose operation only occurs with matrices and vectors. In a scalar there is no transpose operation because it only consists of one row and one column. This causes the scalar value to be equal to the transpose of the scalar.
Transpose matrix has several properties, including the following.
- (AT) T = A
- (A + B) T = A T + B T
- (A – B) T = A T – B T
- (kA) T = kAT where k is a constant
- (AB) T = B T A T
In order to better understand the transpose matrix, You can listen to the example below.
2. Matrix Determinants
The determinant is a value that is calculated through the elements of a matrix similar in shape to a square. The symbol for the determinant of matrix A is det (A), det A, or |A|. A square matrix is a matrix that has the same number of rows and columns.
If the number of rows and columns is different then the determinant cannot be found. Keep in mind that the basic theory of matrices is the addition of columns in a table or subtracting, multiplying, or dividing the values in a column.
The determinant has certain distinctive properties such as in a matrix A and B of order n x n as follows.
- |AB| = |A| |B|
- |AT| = |A|. The symbol T is a matrix transpose.
- |A-1| = 1/|A| or also known as matrix inverse.
- |kA| = kn|A|. K is a real number and n is the order of matrix A.
- If in a matrix all elements, both rows and columns, are 0, then the determinant value is also 0.
- If the two rows or columns in the matrix are the same or multiples, then the determinant value is 0.
In order to better understand the determinant of matrices, You can listen to the following examples of determinants.
a. Determine the determinant value of the matrix of order 2 x 2 below.
b. Determine the determinant value of the matrix of order 3 x 3 below.
sec(A) = 1.1.2 + 2.4.3 + 3.2.1 – 3.1.3 – 1.4.1 – 2.2.2
= 2 + 24 + 6 – 9 – 4 – 8
3. Inverse Matrix
Inverse means the opposite. Meanwhile, the inverse matrix is the opposite of a matrix. If the matrix is multiplied by the inverse it will become an identity matrix. The matrix inverse is denoted by A -1 . The terms of the matrix inverse, namely the determinant value of the matrix is not equal to zero.
Determination of the inverse of a matrix has two rules or ways based on the order. The following details how to determine the inverse.
a. Inverse Matrix Based on Order 2 x 2
The inverse matrix of order 2 x 2 can be found in the following way.
To better understand the inverse matrix of order 2 x 2, You can listen to the following problem.
b. Inverse Matrix Based on Order 3 x 3
The inverse matrix of order 3 x 3 can be found by the Gauss Jordan elimination method. The system can be stated as follows.
The square matrix A is eliminated by means of algebraic operations to form an identity matrix. If matrix A has become an identity matrix, it will turn into the inverse of matrix A.
To better understand the inverse matrix of order 3 x 3, You can listen to the following problem.
Example of a Matrix Problem and its Solution
To better understand the matrix, You can listen to some of the matrix problems and their solutions below.
1. Determine the value of x + y from the matrix below.
When you know an equation in the matrix, what you can do is solve it step by step.
From the matrix operations and matrix similarities above, several equations can be found between them as follows.
So the value of x + y is 23.
2. It is known that matrices A and B are as follows.
If A + B = C, then determine the inverse matrix C!
3. The matrices A and B are known as below.
If C = AB, then determine the value of the inverse matrix C!
4. The travel agent “Lombok Charming” offers travel packages as shown in the table below.
|Package I||Package II|
The appropriate matrix form for determining the cost of renting a hotel each night and the cost of one tourist attraction is…
For example, hotel rental = x and tourist attractions = y, then the table above can be presented in matrix form, more or less like the following.
To get x and y values in the matrix equation, you can use the matrix inverse as follows.
5. Determine the determinant of matrix (A + B) 2 from matrix A and matrix B as follows.
By applying the rules of matrix multiplication and matrix determinant |A n |=|A| n will then be obtained.
So, the determinant of matrix (A + B) 2 is 0.
6. If matrix A = then the value of A 2 – 2A + I is…
By using the matrix multiplication rule can be obtained as follows.
7. The inverse matrix A is known as follows.
How many x matrices satisfy the relationship:
By applying one of the properties of matrix A . A -1 = I so it can be written as follows.
8. Determine the determinant of A T A + BB T from the following two matrices.
So, the value of A T A + BB T = 5
9. Determine matrix A from the product of the following matrices.
10. Determine the inverse of matrix X that satisfies the following equation.
By applying the properties of the matrix A . B = C then B = A -1 . C, then the following equation is obtained.