Singular Matrix: Concepts, Formulas, and Example Problems

When Sinaumed’s was still in high school or studying science that intersected with the calculation of many variables. So, it would be better to study the matrix to make these jobs easier.

There are various types of matrices, one of which is a singular matrix. This matrix has the opposite, which is a non-singular matrix. The singular matrix itself is a matrix that has no inverse and zero determinant.

To understand more about singular matrices, Sinaumed’s can listen to the following explanation about singular matrices.

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.

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.

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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.

The matrix is ​​taught in high school in mathematics. The following is a book recommendation that summarizes high school learning materials other than matrices. Sinaumed’s can use this book to support learning mathematics.

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).

Concept and Characteristics of Singular Matrix

In a book entitled Preparation for the Mathematics Olympiad for Middle/Mts & Equivalent Levels written by I Gusti Agung Oka Yadnya, he formulates that a singular matrix is ​​a matrix whose determinant value is zero (0). The singular matrix has no matrix inverse or inverse matrix.

Singular matrices have the opposite, namely non-singular matrices. The matrix, the determinant is not equal to zero and has an inverse. The inverse itself is an inverse of the two matrices. If the matrix is ​​multiplied it will produce a square matrix (AB = BA = |).

The symbol for the inverse matrix is ​​written with a power of (-1). For example, the inverse matrix A is denoted by A -1 .

To better understand the singular matrix, Sinaumed’s can rely on the following characteristics of the singular matrix.

  • All elements in a row or column are equal to zero.
  • All elements in a row are multiples of elements or members in other rows.
  • All elements in a column are the sum of several other columns.
  • All elements in a column are multiples of elements or members in other columns.
  • All elements in a row are the sum of several other rows.

A matrix can be classified as a singular matrix when it fulfills the following conditions.

  • A matrix of type n x m with a determinant of 0.
  • The submatrix determinant is 0.
  • The determinant of one of the rows or columns is 0.
  • Not all elements in each row and column are 0.
  • It only has one solution because only by calculating the determinant can it be seen that the matrix is ​​singular or non-singular.

Singular Matrix Formula

The principle of the singular matrix is ​​that its determinant is equal to zero. So to determine whether a matrix is ​​included in the singular matrix or not, the determinant must be calculated. The following proves the singular matrix on matrices of order 2 x 2 and 3 x 3.

1. Matrix of Order 2 x 2

0 = (5 x 18) – (6 x 15)

0 = 90 – 90

0 = 0

So, the matrix is ​​included in the singular matrix.

2. Matrix of Order 3 x 3

0 = ((-3) x 1 x 2) + (1 x 5 x 9) + (2 x (-6) x (-7)) – (2 x 4 x 9) – ((-3) x 5 x (-7)) – (1 x (-6) x (-8))

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0 = 96 + 45 + 84 – 72 – 105 – 48

0 = 0

The singular matrix has special properties in its construction. Suppose the matrix A nxn as follows.

A matrix fulfills the special property if and only if it satisfies the following properties.

In general, the above equation can be written as follows.

The matrix is ​​taught in high school in mathematics. The following is a book recommendation that summarizes high school learning materials other than matrices. Sinaumed’s can use this book to support learning mathematics.

Example of a Singular Matrix Problem

In order to better understand the singular matrix, Sinaumed’s can listen to some of the questions and discussions below which have been summarized from various sources.

1. It is known that matrix A fulfills special properties.

Prove that matrix A is a singular matrix!

Answer:

= (12 x 24 x 33) + (22 x 27 x 20) + (25 x 14 x 30) – (25 x 24 x 20) – (12 x 27 x 30) – (22 x 14 x 33)

= 9,504 + 11,880 + 10,500 – 12,000 – 9,720 – 10,164

= 0

2. Determine whether matrix A and matrix B below are singular matrices or not!

Answer:

= ((-1) x 5) – ((-1) x 4)

= -5 + 4

= -1

= (P x (-Q)) – (Q x (-P))

= -PQ + PQ

= 0

So, matrix A is included in the non-singular matrix and matrix B is included in the singular matrix.

3. Determine the value of n for the matrix to become a singular matrix.

Answer:

sec (D) = 0

(n (n + 1)) – ((n + 1) (-4)) = 0

(n 2 + n) – (-4n – 4) = 0

n 2 + n + 4n + 4 = 0

n2 + 5n + 4 = 0

(n + 1) (n + 4) = 0

n + 1 = 0 n = -1

n + 4 = 0 n = -4

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4. If matrix C is singular then calculate the value of x

Answer:

We assume that matrix C is singular, so the determinant of C is 0

0 = ( x ( x + 4)) – (1 x 5)

0 = x 2 + 4 x – 5

0 = ( x – 1) ( x + 5)

x – 1 = 0         x = 1

x + 5 = 0         x = -5

So, the value of x = – 5 or x = 1

5. Prove whether the matrix A below is included in the singular matrix!

Answer:

sec (A) = (-6 x 2) – (-4 x 3)

= -12 – (-12)

= 0

So matrix A is proven to be a singular matrix.

6. Prove if matrix B is included in the singular matrix

Answer:

sec (B) = (6 x (-3)) – (3 x (-6))

= -18 + 18

= 0

So, matrix B is proven to be a singular matrix

7. Calculate the value of t if it is known that matrix A is a singular matrix!

Answer:

Matrix A singular = determinant 0

0 = (5 x 4 x 2) + (2 x 7 x (-5)) + (-3 x (-10) xt) – (-3 x 4 x (-5)) – (5 x 7 xt) – (2 x (-10) x 2)

0 = 40 + (-75) + 30t – 60 – 35t – (-40)

0 = -5t – 50

5t = -50

t = -10

So, the value of t = -10

8. Prove that matrix B is included in the singular matrix!

Answer:

= (0 x 1 x 6) + (-1 x (-7) x 1) + (-1 x (-8) x 5) – (-1 x 1 x 1) – (0 x (-7) x 5) – (-1 x (-8) x 6)

= 0 +7 + 40 + 1 – 0 -48

= 0

So, it is proven that the determinant value of matrix B is equal to 0, so it is included in the singular matrix.