The Concept of Identity Matrix and Example Problems

Matrix is ​​one of the easiest ways to solve mathematical problems that have many variables. Usually it is applied to the fields of mathematics, economics, and engineering.

The matrix itself has many types. One of them is the identity matrix in which the diagonal is the number one. Then, how to operate it. The following will explain in more detail about the identity matrix.

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, Sinaumed’s 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, Sinaumed’s 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.

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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).

The Concept and Properties of the Identity Matrix

The identity matrix is ​​a matrix that has a value on the main diagonal elements in the form of one and the elements outside the main diagonal are zero. The identity matrix has the same properties as the number 1. Which, when a matrix is ​​multiplied by the identity matrix, the result will not change.

The identity matrix is ​​denoted by In or simply I if the size of n can be known from the context of the discussion. Some books use the abbreviations U and E based on “unit matrix” which in Indonesian means unit matrix. The German translation is “einheitsmatrix”.

In order for Sinaumed’s to better understand the identity matrix, you can listen to the example below.

Example Questions and Discussion of the Matrix

To better understand the identity matrix, Sinaumed’s can listen to several examples of identity matrix problems and their solutions.

1. If the matrices A and B are known as follows.

Determine whether AB = A and BA = A.

Answer:

So, it is proven that if AB = BA = A then matrix B is identity matrix I.

2. Prove if A I 3 = A = I 3 A in the following matrices.

Answer:

So, A I 3 = A = I 3 A is proven to be true.

3. If the matrices G and A are known as follows. Then, prove that G + A = A + G = G

Answer:

So, G + A = A + G = G proves to be true.

4. If the matrix is ​​known as below.

Then prove that P + A = A + P = P.

Answer:

So, the matrix operation P + A = A + P = P is proven to be true.

5. Prove if the matrix AB = BA = A in the following matrices A and B.

Answer:

So, AB = BA = A is proven true.

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Types of Matrix Based on Its Constituent Elements

There are various types of matrices. Here are some classifications of matrices based on their constituent elements.

1. Null Matrix

The zero matrix is ​​a matrix in which all the constituent elements are zeros. Here’s an example of a zero matrix.

2. Diagonal Matrix

A diagonal matrix is ​​a matrix in which all elements outside the main diagonal are zero and at least one element on the main diagonal is non-zero. The following is an example of a diagonal matrix.

3. Scalar Matrix

A scalar matrix is ​​a matrix in which all the elements are on the same diagonal. Here’s an example of a scalar matrix.

4. Symmetry Matrix

A symmetric matrix is ​​a square matrix in which every element other than the diagonal elements consists of numbers that are symmetrical about the main diagonal. Here’s an example of a symmetric matrix.

5. The Oblique Symmetry Matrix

A skewed symmetric matrix is ​​a symmetric matrix whose elements, apart from the diagonal elements, are opposite to each other. The following is an example of an oblique symmetric matrix.

6. Identity Matrix

The identity matrix is ​​a matrix in which all the main diagonal elements are one and the non-main diagonal elements are zero. Here’s an example of an identity matrix.

7. Upper Triangular Matrix

The upper triangular matrix is ​​a diagonal matrix whose elements on the right (top) of the main diagonal are not equal to zero. Here’s an example of an upper triangular matrix.

8. Lower Triangular Matrix

The lower triangular matrix is ​​a diagonal matrix in which the elements on the left (bottom) of the main diagonal are not equal to zero. The following is an example of a lower triangular matrix.

9. Transpose Matrix

The transpose matrix is ​​a matrix obtained by transferring row elements to column elements or vice versa. Here’s an example of transpose matrix.

Matrix Type Based on Order

The following is a matrix classification based on the order into several categories as follows.

1. Square/Square Matrix

A square matrix is ​​a matrix of order n x n or the number of rows is equal to the number of columns. Here’s an example of a square matrix.

2. Row Matrix

A row matrix is ​​a matrix that has an order of 1 x n or consists of only one row. The following is an example of a row matrix.

3. Column Matrix

A column matrix is ​​a matrix that only consists of one column or has the order n x 1. The following is an example of a column matrix.

4. Upright Matrix

An upright matrix is ​​a matrix that has the order m x n with m > n. The following is an example of a straight matrix.

5. Flat Matrix

Flat matrix is ​​a matrix of the order m x n with m < n. Here’s an example of a flat matrix.

Matrix Classification Based on the Nature of Operations

Matrices are classified according to the nature of their operations into two categories. Here are the details of both.

1. Singular matrix ( singular matrix )

A singular matrix is ​​a matrix that has a determinant value of zero and has no inverse. Here’s an example of a singular matrix.

2. Non singular matrix ( non singular matrix )

A non-singular matrix is ​​a matrix whose determinant is not equal to zero and has an inverse. The following is an example of a non-singular matrix.