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
A
B
=
B
A
=
I
n
{\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).
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?
$\tiny{311.2.2.6}$
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 $A^{-1}$
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 .
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...
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...
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...
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 =...
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...
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...
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...
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"...
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...
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!
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...
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.
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...
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...
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...
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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?
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...
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...
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...