Content Strategy Manager
Inverse matrix means that the matrix is unique and there is only one unique inverse matrix of that particular matrix. A matrix is an array of numbers that create rows and columns. The number of rows and columns is called dimensions. It is given by m x n where n and m represent the number of columns and rows respectively. A matrix with the name ‘A’ and with dimensions n x n can be inverted if another matrix B exists such that AB = BA = I, where I is the identity matrix.
AB = BA = I. Hence A-1 = B and B is the inverse of A. So, A can also be the inverse of B or B-1 = A.
The matrix can be inverted by using methods like Newton's Method, Cayley-Hamilton Method, Gaussian Elimination or Eigen Decomposition Method.
| Table of Content |
Key Words: Proof, Matrix, Inverse, Invertible, Theorem, Uniqueness, Equation, Identity matrix, division, multiplication, addition, subtraction
What is Matrix?
[Click Here for Sample Questions]
An array of numbers, positioned in such a way that they create rows and columns, is known as Matrix. Dimensions are the number of rows and columns of a matrix. It is given by m x n where n and m represent the number of columns and rows respectively. Matrices are used for common mathematical actions such as division, multiplication, addition or subtraction. Matrices can be classified into a real or a complex matrix depending on whether or not their numbers are real or complex respectively.
What is Inverse Matrix?
[Click Here for Sample Questions]
A matrix with the name ‘A’ and with dimensions n x n can be invertible if another matrix B exists with the same dimension so that AB = BA = I, where I is the identity matrix of the same order. Matrix B is the inverse of matrix A. Inverse matrix A is represented by A-1. An Invertible matrix can also be called a nondegenerate matrix or a non-singular matrix.
Matrix A
Matrix B
A is multiplied with B to obtain an identity matrix:
B is also multiplied by A to obtain the same identity matrix:
AB = BA = I. Hence A-1 = B and B is the inverse of A. So, A can also be the inverse of B or B-1 = A.
An un-invertible square matrix is called degenerate or singular. If the value of the matrix’s determinant is zero, then it’s known as a Singular Matrix.
Also Read:
| Related Articles | ||
|---|---|---|
| Properties of Determinants | Determinant of a Matrix | Application of Matrix and Determinant |
Methods of Inverting Matrix
[Click Here for Sample Questions]
The matrix can be inverted by using the following methods:
- Newton’s Method
- Cayley-Hamilton Method
- Gaussian Elimination
- Eigen Decomposition Method
Also read: Orthogonal Matrix
Theorem of Inverse Matrix
[Click Here for Sample Questions]
The inverse matrix of, say ‘A’ means that the matrix is unique and there is only one unique inverse matrix of A.
That’s why the inverse matrix of A is denoted by A−1.
So to prove the uniqueness, you have two inverse matrices B and C and now prove that fact B=C.
As mentioned above, B is the inverse matrix if it satisfies AB=BA=I,
Proof of the Uniqueness of Inverse Matrix
[Click Here for Sample Questions]
Suppose that there are two inverse matrices B and C of matrix A. Then they satisfy AB=BA=I and AC=CA=I.
To show the uniqueness of the inverse matrix, we show that B=C is as follows. Let I be the
n×n identity matrix.
We have B=BI
=B(AC) by (AC=CA=I)
=(BA)C by associativity
=IC by AB=BA=1
=C.
Thus, we must have B=C, and there is only one inverse matrix of A.
Also Read:
| Related Articles | ||
|---|---|---|
| Matrix Operations | Transpose of a Matrix | Symmetric Matrix |
| Skew Matrices | Operations on Matrices | Determinants |
Example of Invertible Matrix and its Inverse
[Click Here for Sample Questions]
Example: If A=[−3150] and B=[015135], then show that A is an invertible matrix and B is its inverse.
Solution:
We know, A=[−3150] and B=[015135]
Now, find A's determinant.
|A|=|−3150|
= -3(0) – 1(5)
= 0 – 5
= -5 ≠ 0
Thus, A is an invertible matrix.
We know that, if A is invertible and B is its inverse, then AB = BA = I, where I is an identity matrix.
AB = BA = I
Therefore, matrix A is invertible and matrix B is its inverse.
Things To Remember
- An array of numbers, positioned in such a way that they create rows and columns, is known as a Matrix.
- Dimensions are the number of rows and columns of a matrix.
- It is given by m x n where n and m represent the number of columns and rows respectively.
- Matrices are used for common mathematical actions such as division, multiplication, addition or subtraction.
- Matrices can be classified into a real or a complex matrix depending on whether or not their numbers are real or complex respectively.
- A matrix with the name ‘A’ and with dimensions n x n can be invertible if another matrix B exists with the same dimension so that AB = BA = I, where I is the identity matrix of the same order.
- Matrix B is the inverse of matrix A.
- The inverse of matrix A is represented by A-1.
- An Invertible matrix can also be called a nondegenerate matrix or a non-singular matrix.
- The inverse matrix of, say ‘A’ means that the matrix is unique and there is only one unique inverse matrix of A.
- That’s why the inverse matrix of A is denoted by A−1.
- So to prove the uniqueness, you have two inverse matrices B and C and now prove that fact B=C.
- As mentioned above, B is the inverse matrix if it satisfies AB=BA=I,
Sample Questions
Ques. Could someone explain this proof of unique inverse matrices? (3 marks)
Ans. What you've got written isn't evidence. Evidence has phrases explaining what is going on. What is written here is an again-of-the-envelope calculation that would be fleshed out and reorganized into evidence.
A extra whole evidence would be along the strains of:
Suppose that A has inverses B and C. To show that A has a unique inverse, we ought to show that B = C. Now
B = BI = B(AC) = (BA)C = IC = C,
(using the associativity of matrix multiplication in the center step) so B = C and the inverse of A is unique.
Ques. Can the inverse of the matrix have two different answers? (5 marks)
Ans. It cannot. When a matrix is invertible, it has a unique inverse. A very simple proof is as follows:
Let B and C be inverses of an invertible matrix, A (and let I denote the identity matrix of the identical order as those matrices). We will display that B=C, B=BI=B(AC), wherein AC=I because
C is an inverse of A. Now, the use of associativity of matrix multiplication above implies that
B=(BA)C=IC=C wherein BA=I due to the fact B is an inverse of A. Thus, we've proven that
B=C, or in other words, A has a unique inverse.
We have not used the "structure" of matrices in this evidence. All we wished changed into the life of an inverse, a unique identification [which, by the way, has a similar proof], and associativity. Indeed, the proof is generalizable to a far larger context (see: Group Theory).
A non-invertible matrix has a so-referred to as generalised inverse or pseudoinverse, and this isn't always specific. Of course, the generalised inverse is surely widespread, so that a unique case is the generalised inverse of an invertible matrix, which, it turns out, is the same old inverse and is consequently particular.
Ques. What are the minimal requirements for a (rectangular) matrix A, ensuring the existence of (ATA)−1? (2 marks)
Ans. Let A be an m×n matrix. For the matrix ATA to have an inverse, A have to-have rank
- In other words, A must-have complete column rank.
A generalised inverse of a matrix usually exists. In truth, one of the generalized inverses, the Moore-Penrose pseudoinverse, is guaranteed to exist. Not most effective that, but the Moore-Penrose pseudoinverse is precise for every matrix.
Ques. How can you find the inverse matrix with Cramer's rule? (2 marks)
Ans. Cramer’s rule to discover an inverse takes O(n!) operations in case you do it the stupid way and O(n4) operations at first-class (calculate n+1 determinants of size n x n). If you want the inverse (maximum of the time you don’t, you’re better off with the LU decomposition), use Gauss Jordan elimination. That is still highly-priced, but handiest O(n3).
So why does this rule exist you would possibly ask. It has a few theoretical costs because it suggests an important and sufficient condition for the life of a solution, viz det(A)≠0.
Ques. What are the properties of an inverse matrix? (3 marks)
Ans. The homes of an inverse matrix are as follows:
- (A−1)−1 = A
- (kA)−1 = ok−1A−1 for any nonzero scalar ok
- (Ax)+ = x+A−1 if A has orthonormal columns, wherein + denotes the Moore–Penrose inverse and x is a vector
- (AT)−1 = (A−1)T
- For any invertible n x n matrices A and B, (AB)−1 = B−1A−1. More particularly, if A1, A2…, Ak are invertible n x n matrices, then (A1A2⋅⋅⋅Ak-1Ak)−1 = A−1kA−1k−1?A−12A−11
- det A−1 = (det A)−1
Ques. What are the applications of an inverse matrix? (2 marks)
Ans. The applications of a matrix are:
- Least-squares or Regression
- Simulations
- MIMO Wireless Communications
Ques. What does the second theorem of inverse matrix say? (1 mark)
Ans. The second theorem of the inverse matrix says that, If A and B are matrices of the same order and are invertible, then (AB)-1 = B-1 A-1.
Ques. Prove the second theorem of inverse Matrix. (3 marks)
Ans. Proof is as follows:
(AB)(AB)-1 = I (From the definition of inverse of a matrix)
A-1 (AB)(AB)-1 = A-1 I (Multiplying A-1 on both sides)
(A-1 A) B (AB)-1 = A-1 (A-1 I = A-1 )
I B (AB)-1 = A-1
B (AB)-1 = A-1
B-1 B (AB)-1 = B-1 A-1
I (AB)-1 = B-1 A-1
(AB)-1 = B-1 A-
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Check-Out:
Comments