What is Inverse matrix: Definition and 40 Discussions

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate), if there exists an n-by-n square matrix B such that







{\displaystyle \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n}\ }
where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.
A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (m-by-n matrices for which m ≠ n) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n (n ≤ m), then A has a left inverse, an n-by-m matrix B such that BA = In. If A has rank m (m ≤ n), then it has a right inverse, an n-by-m matrix B such that AB = Im.
While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any ring. However, in the case of the ring being commutative, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.
The set of n × n invertible matrices together with the operation of matrix multiplication (and entries from ring R) form a group, the general linear group of degree n, denoted GLn(R).

View More On Wikipedia.org
  1. jv07cs

    I Do Metric Tensors Always Have Inverses?

    I am reading about musical isomorphisms and for the demonstration of the index raising operation from the sharp isomorphism, we have to multiply the equation by the inverse matrix of the metric. Can we assume that this inverse always exists? If so, how could I prove it?
  2. karush

    MHB 311.2.2.6 use inverse matrix to solve system of equations

    $\tiny{311.2.2.6}$ Use the inverse to solve the system $\begin{array}{rrrrr} 7x_1&+3x_2&=-9\\ -2x_1&+x_2&=10 \end{array}$ the thing I could not get here without a calculator is $A^{-1}$
  3. karush

    MHB Calculating the Inverse Matrix for a 3x3 Matrix

    $\tiny{311.2.2.31}$ $A=\left[\begin{array}{rrrrr} 1&0&-2\\-3&1&4\\2&-3&4 \end{array}\right]$ RREF with augmented matrix $\left[ \begin{array}{ccc|ccc} 1&0&-2&1&0&0 \\&&&\\-3&1&4&0&1&0 \\&&&\\ 2&-3&4&0&0&1\end{array}\right] \sim \left[ \begin{array}{ccc|ccc}1&0&0&8&3&1 \\&&&\\0&1&0&10&4&1 \\&&&\\...
  4. patric44

    Efficient Solution for Dividing Matrices: B/A Calculation Explained

    he is asking for the division of the two matrices , so i tried to get the inverse of the matrix A but it appears to get more complex as the delta for A is somehow a big equation . and what really bothers me that there is another A , B inside the matrix B ?! find B/A .
  5. U

    A Solve Badly Scaled Problem with Newton-Chord Technique

    Hello guys, I try to use the Newton - Chord technique in order to solve a nonlinear system and find it's equilibrium points.This method requires the inverse of the Jacobian Matrix of the nonlinear system. After the linearization around the given starting point x0, I create a linear...
  6. M

    MHB Calculation of the inverse matrix - Number of operations

    Hey! :o Let A be a regular ($n\times n$)-Matrix, for which the Gauss algorithm is possible. If we choose as the right side $b$ the unit vectors $$e^{(1)}=(1, 0, \ldots , 0)^T, \ldots , e^{(n)}=(0, \ldots , 0, 1 )^T$$ and calculate the corresponding solutions $x^{(1)}, \ldots , x^{(n)}$ then...
  7. MrsM

    Using eigenvalues to get determinant of an inverse matrix

    Homework Statement Homework Equations determinant is the product of the eigenvalues... so -1.1*2.3 = -2.53 det(a−1) = 1 / det(A), = (1/-2.53) =-.3952 The Attempt at a Solution If it's asking for a quality of its inverse, it must be invertible. I did what I showed above, but my answer was...
  8. D

    Find inverse matrix using determinants and adjoints

    Hello! Please, help me to see my mistake - for quite a while I can't solve a very easy matrix. I have to find the inverse of the given matrix using their determinants and adjoints. 4 6 -3 3 4 -3 1 2 6 to find adjoint matrix I need to find cofactors 11, 12, etc till 33. Cofactor11 =...
  9. ognik

    MHB Uniqueness of Inverse Matrices: Proof and Explanation

    I have an exercise which says to show that for vectors, $ A \cdot A^{-1} = A^{-1} \cdot A = I $ does NOT define $ A^{-1}$ uniquely. But, let's assume there are at least 2 of $ A^{-1} = B, C$ Then $ A \cdot B = I = A \cdot C , \therefore BAB = BAC, \therefore B=C$, therefore $ A^{-1}$ is...
  10. B

    Relationship between inverse matrix and inertia tensor?

    Seems exist some relationship between the inverse of a matrix with the inertia tensor, looks: This relationship really exist?
  11. S

    Linear algebra -- compute the following without finding invA

    Homework Statement Homework Equations A=LU, U^-1 * L^-1= A^-1 , U^-1 * L^-1 * U^-1 * L^-1 = A^-2, The Attempt at a Solution I used MATLAB and the relations: U^-1 * L^-1= A^-1 , U^-1 * L^-1 * U^-1 * L^-1 = A^-2, to find a solution I found U^-1*L^-1 , let =B...
  12. M

    Which Matrix Formulas Are Universally True for Invertible Matrices?

    Homework Statement Determine which of the formulas hold for all invertible nhttp://msr02.math.mcgill.ca/webwork2_files/jsMath/fonts/cmsy10/alpha/144/char02.png n matrices A andB A. 7A is invertible B. ABA^−1=B C. A+B is invertible D. (A+B)2=A2+B2+2AB E. (A+A^−1)^8=A8+A−8 F...
  13. I

    Inverse matrix word problem, matrix arithmetic

    Homework Statement Hello! Please, take a look at the attached picture - there is a quote of the exercise and below is my attempt to make a matrix. Is my matrix correct? I have tried many times to convert it to inverse one, but I can't figure out how to do it - I keep getting "inconvenient"...
  14. S

    Inverse matrix notation question

    I'm hoping that you can help me settle an argument. For a matrix \textbf{M} with elements m_{ij}, is there any sitaution where the notation (M_{ij})^{-1} could be correctly interpreted as a matrix with elements 1/m_{ij}? Personally I interpret (M_{ij})^{-1} in the usual sense of an inverse...
  15. genxium

    Call for help in finding approximate inverse matrix

    I'm looking for solutions to this problem: Matrices A(m,n) and B(n,m) satisfy AB=I(m,m) where n isn't equal to m. Can I find a matrix S(m,n) such that SA=I(n,n) or SA approximates I(n,n)? By approximate I don't have preferred definition, hence any suggestion is welcome!
  16. Y

    MHB Finding element of inverse matrix

    Hello all, I have this matrix A \[A=\begin{pmatrix} 1 &2 &3 &4 \\ 9 &8 &2 &0 \\ 17 &2 &0 &0 \\ 1 &0 &0 &0 \end{pmatrix}\] B is defined as the inverse of A. I need to find the element in the first row and fourth column of B, without using determinants, so without using adjoint. How should I...
  17. Z

    MHB Inverse matrix by row reduction

    Hi, Can anyone help me to inverse the below matrix by row reduction method. I know determinant method but I don't know row reduction method please help me. 4 5 -2 6 thanks.
  18. D

    Finding the inverse matrix responsible for base change in the Z3 Group

    Homework Statement Hey guys, So I have the following permutations, which are a subgroup of S3: σ_{1}=(1)(2)(3), σ_{5}=(1,2,3), σ_{6}=(1,3,2) This is isomorphic to Z3, which can be written as {1,ω,ω^{2}} Next, we have the basis for the subgroup of S3: e_{i}=e_{1},e_{2},e_{3} And we also have...
  19. E

    Inverse matrix mass computation

    Hi ! I've been thinking this problem a whole and I could not find an answer. I want to solve the following problem: suppose I have N mass particles with absolute coordinates \mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N . Besides, I have the following contraints: for all i=1,2,\ldots,N-1...
  20. N

    Matrix and inverse matrix question?

    Homework Statement how do I solve this question https://www.dropbox.com/s/bn958xnm5483s6u/photo%20%281%29.JPG (This link if the image is not visible : https://www.dropbox.com/s/bn958xnm5483s6u/photo%20%281%29.JPG) just so the equation is not clear it says A2 = λA - 2I The inverse should be...
  21. U

    Finding A in Non-Square Matrices: Pseudo Inverse vs Gradient Search

    Hi everybody I have a question that I have a guess for the answer but I want to be sure I have an identity C=A*B where A,B,C are matrices and in general they are not square is there a way to find A in terms of B and C by using pseudo inverse matrix or using pseudo inverse might help me in...
  22. matqkks

    Inverse Matrix: Real-Life Applications & Uses

    What use is the inverse matrix? I would not use it to solve linear systems but there must be some concrete or real life applications where it is used.
  23. P

    Inverse matrix with whole numbers

    Hello, how to find matrix 4x4 which only contains whole numbers and has inverse matrix with whole numbers only aswell? Is there algorithm to find such matrix of n*n? Thanks.
  24. T

    Finding the inverse matrix of fourier transform

    Homework Statement If y=(1,0,0,0) and F4*c=y, find c. Homework Equations c=F4-1*y The Attempt at a Solution I'm stuck. I don't know how to get F4-1. F4-1 = (1/N) * [1, 1, 1, 1; 1 -i (-i)^2 (-i)^3; 1 (-i)^2 (-i)^4 (-i)^6; 1 (-i)^3 (-i)^6 (-i)^9] (this...
  25. B

    What is the expression for the inverse matrix of A?

    1. a) Prove the following holds for A A is a matrix [a b, c d] I is identity matrix. A^2 = (a+d)A-(ad-bc)I. b) Assuming ad-bc not equal to 0, use a) to obtain an expression for A^-1. The Attempt at a Solution I proved the first equation, but I'm not seeing where it relates to...
  26. Demon117

    Computing the derivative of an inverse matrix

    Homework Statement If A, B are elements of Mat(n, R) and A is invertible, compute \frac{d}{dt}_{t=0}(A+tB)^{-1} The Attempt at a Solution The derivative will be of the form \frac{d}{dt}(A+tB)^{-1}=-(A+tB)^{-1}\frac{d}{dt}((A+tB))(A+tB)^{-1} but I need to evaluate this at t=0...
  27. S

    Finding Inverse Matrix by Definition

    Hi fellow mathies, So, I'm wondering if the way to find the inverse of a matrix by definition (instead of using a special algorithm/tacking on the identity matrix and reducing, etc), is to multiply the matrix by a variable matrix and have it equal the identity matrix. So, for example...
  28. L

    Matrix Algebra Inverse Matrix Question

    Homework Statement If A is an nxn matrix such that A^3 = 0 (the zero matrix) then (I-A)^-1 = ...? A. not invertible B. I+A^2 C. I-A D. I+A E. I+A+A^2 Homework Equations The Attempt at a Solution I just don't know how to work out what the inverse of (I-A) is if I know A^3... how is this...
  29. D

    Lineal Algebra: Inverse Matrix of Symmetric Matrix

    Homework Statement Hello, I need some help in the fist parts of two lineal algebra problems, specially with algebraic manipulation. I guess that if I rewrite the determinant nicely some terms get canceled and I can write the inverse nicely, but don't know how to do it... Problem 1...
  30. S

    Calculating the Inverse Matrix with Variables: A Non-Singular Case

    Homework Statement A= [P Q R S] Suppose that A and P are non-singular and show that, A^(-1) = [x -P^(-1)*Q*W -W*R*P^(-1) w ] where W= (S-R*P^(-1)*Q)^(-1) and X= P^(-1)*Q*W*R*P^(-1) Hint: First, remember that if you are given a candidate for an...
  31. P

    Inverse Matrix Problem: How to Solve a Partitioned Matrix

    Homework Statement http://img199.imageshack.us/img199/9336/matho.jpg The attempt at a solution I don't really know how to go about solving this problem, since it's a partitioned matrix. If I write it out in its complete 4 * 4 form, it will take a long time to reduce it, and I won't be...
  32. N

    Solving the Inverse Matrix Problem: Constraints and Proof

    Homework Statement Let a and b be fixed constants and t be a variable. For which values of t is the matrix A = [1 1 1 ] [a b t ] [a^2 b^2 t^2 ] is invertible. Also prove that there is no real 5x5 matrix...
  33. U

    Solve system of equations with inverse matrix

    Homework Statement Find the inverse matrix of A, then use this inverse to solve system of equation. A is a given 3 x 3 matrix and the system of equations is 3 equations in 3 unknowns. Homework Equations The Attempt at a Solution I have found the inverse of A using an...
  34. Q

    Inverse matrix using Hotelling Approximation

    Hello all, I am taking a Numeric Methods course this semester and my professor asked us to investigate Harold Hotelling's method( I suppose this would be and approximation) of finding the inverse of a matrix. I have searched for day and have found many cool things linked to Hotteling but...
  35. L

    What is the Determinant of an Invertible 3x3 Matrix?

    Homework Statement Let A be an invertible 3x3 matrix. Suppose it is known that: A = [u v w 3 3 -2 x y z] and that adj(A) = [a 3 b -1 1 2 c -2 d] Find det(A) (answer without any unknown variables) Homework Equations The Attempt at a Solution I found A^(-1) to be equal...
  36. H

    Is there a standard equation for a 4 by 4 inverse?

    Inquiry: Is there a standard equation for a 4 by 4 inverse? I know that one exists for 3 by 3, 2 by 2, but I cannot find one in my text nor in my searches online. I know I could find one by using the Jordan-Gaussian Method. But, I would be more comfortable with knowing a 4 by 4 general...
  37. B

    Finding the Inverse Matrix for a Finite Set Relation R

    Hi . I have this question( discrete math) : How can the matrix for R-1 , the inverse of the relation R, be found from the matrix representing R, when R is a relation a finite set A. How can I do this problem?
  38. P

    Prove help. rank of inverse matrix

    I can't find out how to prove this question. Can anyone help? Let A be an n x m matrix of rank m, n>m. Prove that (A^t)A has the same rank m as A. Where A^t = the transpose of A. I seen someone else have asked the question before and had got the answer. However I can't understand it...
  39. C

    Solving AxB = (B-1A-1)-1: Inverse Matrix Proof

    I'm having a bit of a struggle with my assignment. I'm supposed to find what is x in AxB = (B-1A-1)-1 . I'm stumped at what to do with this. My friend said that x is I (identity matrix), but he is unable to prove it as well. My linear algebra class just recently started doing this topic...
  40. P

    Guassian elimination and Inverse Matrix

    I'm still having trouble with Guassian elimination and finding the Inverse of a Matrix. I tend to get confused with the switching of the rows or factoring out something. Example matrix 1 1 1 | 1 1 1 -2 | 3 2 1 1 | 2 so it's a system of linear equations and I...