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Matrix is an arrangement of numbers in rows and columns. The elements of matrices are the objects that make up the matrix. The order of a matrix is stated as the number of rows divided by the number of columns.
- Inverse of a Matrix is another matrix, which on multiplication with the given matrix gives the multiplicative identity.
- It is used to find the solution of linear equations through the matrix inversion method.
- Determinant is a scalar value or number calculated using a square matrix.
- Matrix Method is used in order to find the solution of the system of equations.
- In order to solve Linear Equations using the Inverse of a Matrix, all of the variables need to be written in the proper order.
| Table of Content |
Key Terms: Matrix, Inverse of a Matrix, Linear Equations, Determinant, Square Matrix, Matrix Inversion, Adjoint
Inverse of a Matrix
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Inverse of a Matrix is another matrix that on multiplication with the given matrix gives the multiplicative identity. Inverse of a Matrix A is denoted by A-1. If A is a non-singular square matrix, then there exists a n x n matrix A-1, often known as the inverse matrix of A, which satisfies the following property:
| AA-1 = A-1A = I |
Where I refer to the Identity Matrix.
Inverse of a Matrix is obtained by dividing the adjoint of a matrix by the determinant of a matrix. It is calculated using the given steps:
- Step 1: First, determine the minor of the provided matrix.
- Step 2: Convert the obtained matrix into the cofactors matrix.
- Step 3: Then adjugate, and multiply it by the determinant’s reciprocal.
The video below explains this:
Matrices Detailed Video Explanation:
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Determinant of a Matrix
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Determinant of a Matrix is the single unique value representation of a matrix.
- It is the scalar value produced for a given square matrix.
- Determinant is used in linear algebra, and it is computed using the elements of a square matrix like 2 × 2, 3 × 3, 4 × 4, etc.
- Assume S is a set of square matrices, R is a set of integers, and f: S → R is defined by f (A) = k, Where A ∈ S and k ∈ R, then f (A) is referred to as Detrminant of A. It is represented by two vertical lines, i.e. |A|.
Determinant Examples
Here are some examples of Determinant of a Matrix
- Determinant of 2 × 2 Matrix:\(\begin{bmatrix} a & c \\[0.3em] b & d \\[0.3em] \end{bmatrix} = ad - bc\)
- Determinant of 3 × 3 Matrix:\(\begin{bmatrix} a & b & c \\[0.3em] d & e & f \\[0.3em] g & h & i \end{bmatrix} = a(ei - fh) - b(di - gf) + c(dh - ge)\)
Determinants and Matrices as Equation Solver
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Inverse of a Matrix is used to find the Solution of a System of Linear Equations. The determinant of a Matrix is used to check for the consistency of linear equations.
- Consistent System: If one or more solution(s) exists for a system of equations, then, it is referred to as a consistent system of linear equations.
- Inconsistent System: If no solution exists for a system of equations, then, it is referred to as an inconsistent system.
Solution of System of Linear Equations Using Inverse of a Matrix
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Solution of System of Linear Equations can easily be calculated using the inverse of a matrix. Consider the following system of equations:
- a11x + a12y + a13z = b1
- a21x + a22y + a23z = b2
- a31x + a32y + a33z = b3
Where
- x, y, and z refer to the variables.
- a11, a12, …, a33 are the respective coefficients of the variables.
- b1, b2, and b3 are the constants.
The solution for the values of the variables in this system of equations needs to be calculated.
Determinant as an Equation Solver
The given system of equations can be expressed in the form of a square matrix as follows:
It means AX = B or,
In this situation, two major cases arise as follows:
Case 1: If A is a non-singular matrix i.e.|A| ≠ 0, then it has an inverse.
We know that A X = B
Or, A–1 (A X) = A–1 B (Pre-multiplying by A–1)
Or, (A–1 A) X = A–1 B
And, I X = A–1 B (I is the identity matrix)
Or, X = A–1 B where A–1 = (adj A) ⁄ |A|
This method is known as Matrix Method and provides a unique solution of the system of linear equations.
Case 2: If A is a singular matrix, i.e. | A| = 0, Calculate (adj A) B.
- If (adj A) B ≠ O, (O being zero matrices), then the solution of the equations doesn’t exist and the system of equations is called inconsistent.
- If (adj A) B = O, then the system may be either a consistent system or an inconsistent system accordingly as the system has either infinitely many solutions or no solution.
Solution of System of Linear Equations Solved Example
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Here is a solved example on Solution of System of Linear Equations Using Inverse of a Matrix:
| Example: There are three numbers whose sum is given as 3. If the second number is multiplied by 2 and the first number is added to it, 6 is obtained. If the third number is multiplied by 4 and the second number is added to it, 10 is obtained. Represent the given situation algebraically and find the numbers using the matrix method. Solution: Consider that x, y, and z represent the first, second, and third numbers, respectively. Thus, according to the question,
It can be written as AX = B where
|A|= 1(8 – 0) – 1(4 – 0) + 1(1 – 0) = 8 – 4 + 1 = 5 ≠ 0. Now, we need to find adj A.
Adj. A will be \(\begin{bmatrix} 8 & -3 &-2 \\[0.3em] -4 & 4 &1 \\[0.3em] 1 & -1 & 1 \end{bmatrix}\) Thus,
X = A-1B
Hence,
|
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Things to Remember
- Matrix is a collection of numbers organized in rows and columns.
- The number of rows divided by the number of columns determines the order of a matrix.
- Inverse of a Matrix is another matrix that yields the multiplicative identity when multiplied by the given matrix.
- Matrix Inversion Method uses the inverse of a matrix to find the solution to linear equations.
- In order to find the inverse of a square matrix, the square matrix must be non-singular and have a determinant value that is not zero.
- If a Matrix A a non-singular matrix i.e. |A| ≠ 0, then its inverse exists.
Previous Year Questions
- If P and Q are symmetric matrices of the same order then… (KCET - 2019)
- If A and B are square matrices of the same order such that… (KCET - 2009)
- If A and B are two square matrices such that AB = A and… (AMUEEE - 2016)
- If A is a square matrix. A’ its transpose, then… (JKCET - 2008)
- How many 3×3 matrices M with entries from {0,1,2} are… (JEE Advanced - 2017)
- The system of linear equations: x+y+z=0… (VITEEE - 2019)
- If A is a square matrix such that… (COMEDK UGET - 2005)
- If the three linear equations x+4ay+az=0…
- A square matrix A=[aij]n×n is called a lower triangular matrix…
- Which one of the following is true always for any two non-singular…
Sample Questions
Ques. Consider a set of three numbers. The sum of the two numbers and the twice of the second number is equal to 2. The sum of the second and third numbers when subtracted from the twice of the first number gives 1. The difference between thrice the first and five times the third gives 5. Write the given statements in form of a system of equations and solve them using Matrix Method. (5 Marks)
Ans. Let x, y, and z be the three numbers. According to the question, the equations will be
- x + 2y + z = 2
- 2x – y – z = 1
- 3x – 5y = 5
In form of a matrix, it will be written as
Determinant of A = |A| = 1(5 – 0) – 2(–10 + 3) + 1(0 + 3) = 22 ≠ 0. Thus, it means that there exists a unique solution for X.
On the calculation of adj (A), we get Aij = (–1)(i + j) Mij, where Mij refers to the co-factor of aij.
- A11 = 1(5 – 0) = 5, A12 = –1(–10 + 3) = 7, A13 = 1(0 + 3) = 3,
- A21 = –1(–10 –0) = 10, A22 = 1(–5 – 3) = –8, A23 = –1(0 – 6) = 6,
- A31 = 1(–2 + 1) = –1, A32 = –1(–1 – 2) = 3, A33 = 1(–1 – 4) = –5
Inverse of a Matrix A is A−1.
As X = A–1 B
Or
Thus, x = 15/22, y = 21/22, and z = -13/22.
Ques. How to find the Inverse of a 3x3 Matrix? (3 Marks)
Ans. To find the inverse of a 3x3 matrix, the following steps need to be followed:
- Calculate the determinant of the given matrix and see if it's invertible.
- Calculate the determinant of the 2 multiplied by 2 (2x2) minor matrices.
- Formulate the cofactor matrix.
- To get the adjugate matrix, transpose the cofactor matrix.
- Finally, multiply the determinant by each term of the adjugate matrix.
Ques. What are the properties of the Inverse of a Matrix? (3 Marks)
Ans. The properties of the Inverse of a Matrix are as follows:
- In order to find the inverse of a square matrix, the square matrix must be non-singular and have a determinant value that is not zero.
- The numbers are represented by a, b, c, and d.
- The determinant of matrix A is expressed as ad-bc, with a value that is not zero.
Ques. What is Inverse of a Matrix? (2 Marks)
Ans. Inverse of a Matrix is another matrix that gives the multiplicative identity on multiplication with the given matrix. For a given matrix A, its inverse is A-1, and
A · A-1 = A-1· A = I
Where I denote the Identity Matrix.
Ques. What are the various types of Matrices? (3 Marks)
Ans. Matrix is defined as a rectangular array of numbers. The different types of matrices are as follows:
- Row Matrix
- Column Matrix
- Zero or Null Matrix
- Singleton Matrix
- Horizontal Matrix
- Vertical Matrix
- Square Matrix
- Diagonal Matrix
- Scalar Matrix
- Identity (Unit) Matrix
- Equal Matrix
- Triangular Matrices
- Singular Matrix
- Non-Singular Matrix
Ques. What is a Determinant? (2 Marks)
Ans. Determinant of a Matrix is defined as the single unique value representation of a given matrix. It can be calculated with reference to any row or column of a given matrix. Determinant of a matrix is equal to the sum of the product of the elements and their cofactors, of a particular row or column of the matrix.
Ques. How to use basic operations to find the Inverse of a Matrix? (3 Marks)
Ans. To find the inverse of a matrix, follow the steps mentioned below:
- Step 1: Make n x n identity matrix.
- Step 2: To make the original matrix(A) identical to the identity matrix, perform row or column operations on it.
- Step 3: Apply the same techniques to the identity matrix.
Ques. What does Order of a Matrix mean? (2 Marks)
Ans. Order of a Matrix is determined by the number of rows and columns. The numbers in the matrix are the entries, and each number is referred to as an element. The size of a matrix is expressed as m x n, where n is the number of rows and m is the number of columns and is referred to as a 'n by m' matrix.
Ques. What is the Inverse of 2 × 3 Matrix? (2 Marks)
Ans. The first condition for the inverse of a matrix to exist is that the given matrix should be a square matrix. Secondly, the determinant of the given square matrix should be non-zero. It indicates that the inverse of matrices of the order m × n will not exist where m ≠ n. Thus, the inverse of a 2 × 3 matrix cannot be calculated.
Ques. What is the Determinant of Inverse Matrix? (3 Marks)
Ans. Determinant of the inverse of an invertible matrix is the inverse of the determinant of the original matrix, i.e. Det(A-1) = 1 / Det(A). It can be explained as follows:
We know that
Det(A • B) = Det (A) × Det(B)
Also, A × A-1 = I
Det(A × A-1) = det(I)
Or, Det(A) × Det(A-1) = Det(I)
As Det(I) = 1
Det(A) × Det(A-1) = 1
Or Det(A-1) = 1/Det(A)
Hence Proved
Ques. What is Cofactor? (1 Mark)
Ans. Cofactor of an element is the value calculated by multiplying the minor with -1 to the exponent of the sum of the row and column elements in order representation of that element.
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