- $$A =\begin{bmatrix} 1 & 0 & 0\\ 0 Â & 1 & 0\\ 0 & 0 & 1\end{bmatrix} Â \begin{bmatrix} 1 & 2 & 0\\ 2 Â & 0 & -1\\ 2 & 3 & -1\end{bmatrix}$$
- A square matrix that is not invertible is called singular or degenerate. Furthermore, the n-by-n invertible matrices are a dense open set in the topological space of all n-by-n matrices
- However, while all invertible matrices are square, not all square matrices are invertible. Always be careful of the order in which you multiply matrices. For instance, if you are given B and C and asked..
- MATLAB Matrix Tutorial: Matrix Multiplication, Definition, and Operation. February 11, 20190Comments. MATLAB For Beginners: 20-Minute Video Training Course
- Invertible matrices are very important in many areas of science. For example, decrypting a coded message uses invertible matrices (see the coding page). The problem of finding the inverse of a..

* The invertible matrix theorem is a theorem in linear algebra which offers a list of equivalent conditions for an nÃ—n square matrix A to have an inverse*. Matrix A is invertible if and only if any (and hence.. If A is a matrix such that inverse of a matrix (Aâ€“1) exists, then to find an inverse of a matrix using elementary row or column operations, write Â A = IA and apply a sequence of row or column operation on A = IA till we get, I = BA. The matrix B will be the inverse matrix of A. By using elementary operations, find the inverse matrix 16 Matrix Inverse Invertible matrices Invertible Matrices - Example Example If A , B , C are invertible, simplify ( A T C - 1 A ) - 1 TERM One '14. TAGS ax, Invertible matrix, Inverse element $$ A B =\begin{bmatrix} 4 Â – Â 3 & -6 + 6\\ 2 Â – 2 & -3 + 4\end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}$$

- AB = BA = I ……….(1) Since C is also the inverse of A AC = CA = I ……….(2) Thus B = BI = B (AC) = (BA) C = IC = C
- The EFE matrix (Extern Factor Evaluations) en de IFE matrix (Internal Factor Evaluations) are analysis methods to evaluate the organisations' strategy
- ant is non-zero. The deter
- An interactive matrix multiplication calculator for educational purposes..
- $$A =\begin{bmatrix} 1 & 0 & 0\\ 0 Â & -1/3 & 1/3\\ 2 & 0 & -1 \end{bmatrix} Â \begin{bmatrix} 1 & 2 & 0\\ 0 Â & 1 & 0\\ 0 & 1 & 1\end{bmatrix}$$

More generally, a matrix is **invertible** if and only if its determinant is not equal to [math]0[/math], which is equivalent to An n x n matrix A is said to be **invertible** if there exists an n x n matrix B such that VDTs (presumably Virtual Delirium Tremens) was a medical condition in the Real World that was speculated to result from withdrawal from the Matrix. The only surviving record of a patient suspected of having VDTs is Bane, who had self-inflicted wounds on his arms, caused by Smith.. In this lesson we will understand the concept of Invertible matrix. We will also understand the concept of Singular and non singular matrix as well The confusion matrix itself is relatively simple to understand, but the related terminology can be I wanted to create a quick reference guide for confusion matrix terminology because I couldn't find an..

Creating MatricesÂ¶. The linear algebra module is designed to be as simple as possible. In addition to creating a matrix from a list of appropriately-sized lists and/or matrices, SymPy also supports more.. ** Inverse of a Matrix Matrix Inverse Multiplicative Inverse of a Matrix**. Note: Not all square matrices have inverses. A square matrix which has an inverse is called invertible or nonsingular, and a..

- ing column vectors in the matrix.
- Invertible matrix is also known as a non-singular matrix or nondegenerate matrix. For example, matrices A and B are given below: Now we multiply A with B and obtain an identity matri
- Invertible Matrices. I am sure we all might have multiplied a number with it's reciprocal to find the value of a particular equation avoiding the division operation for simplicity
- ant, inverse A matrix is a two-dimensional array of values that is often used to represent a linear transformation or a..
- Improve your math knowledge with free questions in Is a matrix invertible? and thousands of other math skills

inverted file. inverter. invertible matrix. invested. to investigate Why not 1/A ? Because we don't divide by a matrix! And anyway 1/8 can also be written 8-1 Matrix Keyboards is a new gaming company started in 2018 driven by the rising demands of high All of us here at Matrix Keyboards have been gaming our whole lives and truly care about the.. A singular matrix is infinitely hard to invert, and so it has infinite condition number. The perturbed matrix may be theoretically invertible but practically non-invertible Conversely, any invertible 3Ã—3 matrix defines a projective transformation of the plane. Projective transformations (if not affine) are not defined on all of the plane, but only on the complement of a line..

Matrix, the one with numbers, arranged with rows and columns, is extremely useful in most scientific fields. The Matrix, Inverse. For matrices there is no such thing as division, you can multiply but.. Reduce the left matrix to row echelon form using elementary row operations for the whole matrix (including the right one). As a result you will get the inverse calculated on the right. If a determinant of.. Clients. To connect to the Matrix federation, you will use a client. These are some of the most popular Matrix clients available today, and more are available at try-matrix-now Determinants of matrices. Domain of a function. Inequality of arithmetic and geometric means. Matrices and systems of equations. Multi-step equations ** Invertible matrix -- an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is**..

invertible-matrix definition: Noun (plural invertible matrices) 1. (linear algebra) A square matrix which, when multiplied by another (in either order), yields the identity matrix Transforming a non-invertible matrix into an invertible matrix. Given a non-invertible, diagonalizable matrix $A$, I wish to transform it into another matrix $B$ that satisfie

- ation.
- (AB) (AB)â€“1Â = 1 Aâ€“1 (AB) (AB)â€“1Â = Aâ€“1 I (Aâ€“1A) B (AB)â€“1Â = Aâ€“1 IB (AB)â€“1Â = Aâ€“1 B (AB)â€“1Â = Aâ€“1 Bâ€“1 B (AB)â€“1Â = Bâ€“1 Aâ€“1 I (AB)â€“1Â Â =Â Bâ€“1 Aâ€“1 Hence (AB)â€“1 = Bâ€“1 Aâ€“1
- Making a pairwise distance matrix in pandas. This is a somewhat specialized problem that forms part of a lot of data science and clustering workflows. It starts with a relatively straightforward question: if we..
- í ¼í¾¦ Matriz invertible. Quite the same Wikipedia. Just better. Matriz invertible. De Wikipedia, la enciclopedia libre
- invertible matrix (plural invertible matrices). (linear algebra) An nÃ—n square matrix for which some other such matrix exists such that when they are multiplied by each other (in either order), the result is the nÃ—n identity matrix

A matrix is a rectangular array of numbers, arranged in rows and columns. For instance, the left matrix has two rows and three Matrix Multiplication. Transpose and Determinant. Inverting Matrices Matrix addition, multiplication, inversion, determinant and rank calculation, transposing, bringing to diagonal, triangular form, exponentiation, solving of systems of linear equations with solution steps

Square matrices A and B are similar if there exists an invertible matrix X such that B = Xâˆ’ 1AX, and similar matrices have the same eigenvalues where A is an invertible square matrix, T represents transpose and. is equal to. . We know that if, we multiply any matrix with its inverse we get Recognizing when a matrix is invertible or not. Invertible and noninvertibles matrices. Khan Academy. Ð—Ð°Ð³Ñ€ÑƒÐ·ÐºÐ°.. 1) Enter the coefficient matrix in the table labeled Matrix A, note that in the right menu you can add rows and columns using the To calculate the inverse of the matrix, click the menu option Invert

In other words: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc).Question 1: If AÂ is a square matrix of order 3Â and det A Â = 5Â then what is detÂ [(2A)âˆ’1]Â equal to? Matrix Algebra: Table of Contents. Introduction. What is a matrix? Types of matrices. This lesson defines the matrix inverse, and shows how to determine whether the inverse of a matrix exists For example, if you had to solve (5/8)Â x = 10, you could still divide both sides by 5/8, but it would probably be easier to multiply both sides by 8/5. The reciprocal fraction 8/5 is the inverse of 5/8 when you multiply the two fractions, you get 1, which is, here, called as “the (multiplicative) identity”. 1 is called the identity because multiplying something by 1 doesn’t change its value. In the proof that a matrix is invertible if and only if it is full-rank, we have shown that the inverse can be constructed column by column by finding the vectors that solve that is, by writing the vectors of the..

Looking for the definition of invertible matrix? 'Matrix' is one option -- get in to view more @ The Web's largest and most authoritative acronyms and abbreviations resource Also note how the rows and columns are swapped over ("Transposed") compared to the previous example. The Ansoff Matrix, also called the Product/Market Expansion Grid, is a tool The matrix shows four strategies that can be used to help a firm grow and also analyzes the risk associated with each strategy Calculations like that (but using much larger matrices) help Engineers design buildings, are used in video games and computer animations to make things look 3-dimensional, and many other places.$$A =\begin{bmatrix} 1 & 2/3 & -2/3 \\ 0 Â & -1/3 & 1/3\\ 2 & 1/3 & -4/3 \end{bmatrix}Â $$

The amsmath package provides commands to typeset matrices with different delimiters. Once you have loaded \usepackage{amsmath} in your preamble, you can use the following environments in.. * where is an matrix, is an matrix, is an matrix, and is an matrix*. If is invertible, then we can manipulate this matrix via block Gaussian elimination as. and on taking determinants using (1).. A matrix is a collection of data elements arranged in a two-dimensional rectangular layout. We reproduce a memory representation of the matrix in R with the matrix function

24-24? That equals 0, and 1/0 is undefined. We cannot go any further! This Matrix has no Inverse.** If the square matrix has invertible matrix or non-singular if and only if its determinant value is Step 1 : Obtain the determinant of the given matrix**. Since det(A) is not equal to zero, A is invertible Well, what if you could find the inverse of A, something similar to finding the reciprocal fraction above? The inverse of A, written as A-1Â and pronounced as A inverse, would allow you to cancel off the A from the matrix equation and then solve for B. Matrix is similar to vector but additionally contains the dimension attribute. All attributes of an object can be checked with the attributes() function (dimension can be checked directly with the dim() function)

The matrix() CSS function defines a homogeneous 2D transformation matrix. The matrix() function is specified with six values. The constant values are implied and not passed as parameters; the other.. If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A and it is denoted by Aâ€“1. For example, letÂ $$A=\begin{bmatrix} 2 & 3\\ 1 & 2\end{bmatrix} and \: B=\begin{bmatrix} 2 & -3\\ -1 & 2\end{bmatrix}$$ be two matrices. DEFINITION The matrix A is invertible if there exists a matrix A 1 such that A 1A D I and AA 1 D I: (1). Not all matrices have inverses. This is the rst question we ask about a square matrix: Is A invertible

I = identity matrix which is the matrix equivalent to 1. Please take account of the fact that not all the square matrices have inverses, thus those having an inverse are called nonsingular or invertible.. ** The identity matrix is a square matrix that has 1's along the main diagonal and 0's for all other One concept studied heavily in mathematics is the concept of invertible matrices, which are those**..

Invertible matrix - Wikipedia, the free encyclopedia. 5.2 Matrix inverses in MIMO wireless communication 6 See also 7 Notes 8 References 9 External links (redirected from Invertible matrix) Also found in: Dictionary, Thesaurus, Wikipedia. The inverse of a nonsingular matrix A is the matrix A -1 where A Â· A -1= A -1Â· A = I, the identity matrix Answer : If A is of order 3 then, A-1 is also of order 3.Â Now, det (cA) = cn (det A) where n is the order of the matrix. And, det A = 1 / det A-1.Â Thus det [(2A)-1] = 23 det[A] and det[A-1] = 1 / 8 . 5 = 1 / 40

Keynote: The above equation should not be invertible. There are two special keywords which we need to understand: Determinant of a matrix and an identity matrix Warnings: [1] Covariance matrix calculated using the outer product of gradients (complex-step). Notice that I also had to disable the warnings for a matrix that isn't invertible in order to even get this.. Alibaba.com offers 535 invertible matrices products. About 0% of these are Digital Signage and Displays. A wide variety of invertible matrices options are available to you, such as certification 2x2 Invertible matrix. We are about to start a series of lessons dedicated to the inverses of matrices. The topic of today is to learn to identify those matrices which can be inverted and those which can't

Reciprocal of a Number. The Inverse of a Matrix is the same idea but we write it A-1. Why not 1/A ? Because we don't divide by a matrix! And anyway 1/8 can also be written 8-1 2.8 The Invertible Matrix Theorem I â€¢ Be able to use the Invertible Matrix Theorem to draw conclusions related to the invertibility of a matrix. True-False Review For Questions 1-4, decide if the.. When a matrix cannot be inverted and the reasons why it cannot be inverted? How to know if a matrix is A singular matrix is one which is non-invertible i.e. there is no multiplicative inverse, B.. Matrix 2 of 5. Identcode. Leitcode After reading this page, don't miss our catalogue of dense matrix decompositions. The solution: You can choose between various decompositions, depending on what your matrix A looks like, and..

- Invertible matrix. This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by..
- (Imagine in our bus and train example that the prices on the train were all exactly 50% higher than the bus: so now we can't figure out any differences between adults and children. There needs to be something to set them apart.)
- Tool to invert a matrix. The inverse of a square matrix M is a matrix denoted M^-1 such as que M.M^-1=I where I is the identity matrix. How to prove that a matrix is invertible

- When we multiply a matrix by its inverse we get the Identity Matrix (which is like "1" for matrices):
- Homework Statement Prove: If A and B are both nxn matrices, A is invertible and AB=BA, then A-1B=BA-1 Homework Equations (AB)-1=B-1A-1 The..
- Online Gematria Calculator with same phrases values search and words. English Gematria, Hebrew Gematria and Jewish Gematria and Numerology..
- Inverses and Elementary Matrices. Matrix inversion gives a method for solving some systems of equations. Suppose. is a system of n linear equations in n variables. Write
- And it makes sense ... look at the numbers: the second row is just double the first row, and does not add any new information.

A square matrix that is not invertible is called singular or degenerate. An alternative is the LU decomposition, which generates upper and lower triangular matrices, which are easier to invert Why are non-square matrices not invertible? I am running into a lot of doubts like this in my introductory study of linear algebra 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. where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication

How to Calculate Confusion Matrix for a 2-class classification problem? Let's understand confusion matrix through math. Recall. Out of all the positive classes, how much we predicted correctly There is another way to create a matrix in python. It is using the numpy matrix() methods. You can also find the dimensional of the matrix using the matrix_variable.shape We just mentioned the "Identity Matrix". It is the matrix equivalent of the number "1": Question 1. If A and B are invertible matrices of order 3, |A| = 2, |(AB) -1 | = - 1/6

More generally, a matrix is invertible if and only if its determinant is not equal to [math]0[/math], which is equivalent to An n x n matrix A is said to be invertible if there exists an n x n matrix B such that Is Every Invertible Matrix Diagonalizable? The zero matrix is a diagonal matrix, and thus it is diagonalizable. However, the zero matrix is not invertible as its determinant is zero Sal shows why a matrix is invertible if and only if its determinant is not 0 An invertible matrix can be inverted to cancel the original matrix in a multiplication, a singular matrix is a matrix that cannot be inverted, and an ill-conditioned matrix is invertible, but can.. Get Invertible Matrix essential facts below. Add Invertible Matrix to your PopFlock.com topic list for future reference or share this resource on social media

**With matrices the order of multiplication usually changes the answer**. Do not assume that AB = BA, it is almost never true. The function calculates the matrix of an affine transform so that The function calculates the following matrix: where. The transformation maps the rotation center to itself

If a 2Ã—2 matrix A is invertible and is multiplied by its inverse (denoted by the symbol Aâˆ’1), the resulting product is the Identity matrix which is denoted by I. To illustrate this concept, see the diagram below $$A =\begin{bmatrix} 1 & 0 & 0\\ 0 Â & -1 & 0\\ 0 & 0 & 1\end{bmatrix} Â \begin{bmatrix} 1 & 2 & 0\\ 0 Â & 3 & 0\\ 2 & 3 & -1\end{bmatrix}$$ Learn about invertible transformations, and understand the relationship between invertible matrices and invertible transformations. Recipes: compute the inverse matrix, solve a linear system by taking..

Pristine Fractal Relics. Integrated Fractal Matrices Consider any matrix which is invertible say 2*2 in which sum of rows is 1, what we get is sum of all elements of A-1 as 2. similarly u can check it for a 3*3 matrix it comes out to be 3

- g a single elementary row operation on an identity matrix
- Characterizations of Invertible Matrices. Exercises. p.115. 2.4. Partitioned Matrices. Exercises. p.121. 2.5. Matrix Factorizations. Exercises. p.129
- Since these matrices does not have an inverse, I wondered if somebody knew of some sort of pseudo-inverse A 3 by 3 skewsymmetric matrix will not be invertible; no matter what you multiply it by, you..
- matrix - Hessian matrix - hologram matrix - idempotent matrix - identity matrix - immittance matrix - impact matrix - impedance matrix - incidence matrix - information matrix - instrumental variables..
- Definition:Invertible Matrix. From ProofWiki. Jump to navigation Jump to search. Contents. 1 Definition. 2 Non-Invertible Matrix. 3 Also known as
- (Uniqueness of inverse) The Inverse of a square matrix, if it exists, is unique. Proof Let A = [aij] be a square matrix of order m. If possible, let B and C be two inverses of A. We shall show that B = C. Since B is the inverse of A
- ant of a rotation matrix is 1. 3. The inverse of a rotation matrix is equal to its transpose

- The [page:Matrix4] class has various methods for modifying the matrix: object.matrix.setRotationFromQuaternion( quaternion ); object.matrix.setPosition( start_position..
- Answer: According to the definition; the inverse of an â€˜nâ€™ x â€˜nâ€™ matrix â€˜Aâ€™ is the â€˜nâ€™ x â€˜nâ€™ matrix â€˜Bâ€™. Just like â€˜ABâ€™ = â€˜BAâ€™ = â€˜Iâ€™. Hence, if we will get the inverse of a 4 x 4 matrix â€˜Aâ€™ to be â€˜Bâ€™, then for checking our work, we will have to simply multiply â€˜ABâ€™ and â€˜BAâ€™.
- dÃ©finition - Invertible_matrix. voir la dÃ©finition de Wikipedia. In linear algebra an n-by-n (square) matrix A is called invertible (some authors use nonsingular or nondegenerate) if there exists an..
- Linear algebra (numpy.linalg). scipy doc. Inverse of a matrix using numpy. stackoverflow. Matrix Inversion: Finding the Inverse of a Matrix
- Convert your given matrices into the reduced row echelon form using Rref calculator in seconds. 1/ To assess whether a matrix is invertible and finally find its reverse: whenever you use the basic..
- List of labels to index the matrix. This may be used to reorder or select a subset of labels. If None is given, those that appear at least once in y_true or y_pred are used in sorted order

Returns a matrix from an array-like object, or from a string of data. A matrix is a specialized 2-D array that retains its 2-D nature through operations. It has certain special operators, such as.. $$A B =\begin{bmatrix} 2 & 3\\ 1 & 2\end{bmatrix}\begin{bmatrix} 2 & -3\\ -1 & 2\end{bmatrix}$$ The product of nxn invertible matrices is invertible, and the inverse is the product of their inverses in the REVERSE order. States theorem 6 Matrices are readonly - methods and operators do create new matrix objects. All methods return readonly data or deep clones - also the constructor does a deep clone of provided data Title:Invertible sums of matrices. Authors:Justin Chen. Abstract: We give an elementary proof of a Caratheodory-type result on the invertibility of a sum of matrices, due first to Facchini and Barioli

using MathNet.Numerics.LinearAlgebra; using MathNet.Numerics.LinearAlgebra.Double; Matrix<double> A = DenseMatrix.OfArray(new double In mathematics and especially linear algebra, an n-by-n (square) matrix A is called invertible, non-singular, or regular if there exists another n-by-n matrix B such that. AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication Matrix Calculator computes all the important aspects of a matrix: determinant, inverse, trace , norm. Just enter the matrix, choose what you want to calculate, push the button and let the matrix..

- Plain matrices are used in special cases like setting up nonstandard camera projection. Matrices in Unity are column major; i.e. the position of a transformation matrix is in the last column, and the first..
- See how Matrix Cloud can accelerate your study and make it paperless. Dmitry Sharov, the founder of Data MATRIX, formed part of Rusbase ranking with the the most successful Biotech entrepreneurs
- This article is showing a geometric and intuitive explanation of the covariance matrix and the way it describes the shape of a data set. We will describe the geometric relationship

To understand inverse calculation better input any example, choose "very detailed solution" option and examine the solution. The matrix is n by n, where n is the number of classes. The simplest classifiers, called binary classifiers, has only two classes: positive/negative, yes/no, male/femal The product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order. False it should be the reverse order, Theorem 6b (2.2) Invertible matrix - Free download as PDF File (.pdf), Text File (.txt) or read online for free. Organization : Saylor.org Instructions for Invertible matrices

Translation matrices. The Identity matrix. Cumulating transformations : the ModelViewProjection matrix. Putting it all together Compressed Sparse Row matrix. This can be instantiated in several ways to construct an empty matrix with shape (M, N) dtype is optional, defaulting to dtype='d'

Matrix Calculator (beta): A beautiful, free matrix calculator from Desmos.com. This Matrix Calculator is in Beta. Send us feedback We learn how to find the inverse of a matrix, whihc we use later to solve systems of linear equations The calculator computes the adjugate matrix of a given NxN matrix and uses the result to compute also The calculator shows the calculation of every element of the adjugate matrix. The input field N.. definition - Invertible matrix. definition of Wikipedia. Advertizing â–¼. In linear algebra an n-by-n (square) matrix A is called invertible (some authors use nonsingular or nondegenerate) if there..

In Linear Algebra, an identity matrix (or unit matrix) of size $n$ is an $n \times n$ square matrix An inverse of a matrix is also known as a reciprocal matrix. Now we pick an example matrix from a.. 4. Determine whether the following matrix is invertible, and if so, nd its inverse We put the original matrix on the left side of an augmented matrix, and put the identity matrix on the right side In that example we were very careful to get the multiplications correct, because with matrices the order of multiplication matters. AB is almost never equal to BA.

In linear algebra, an n-by-n square matrix A is called invertible if there exists an n-by-n square matrix B such that. For faster navigation, this Iframe is preloading the Wikiwand page for Invertible matrix 3.1.10 Invertible Matrices. another square matrix B of the same order m Ã— m, such that AB = BA = Im, then, A is said to be invertible matrix and B is called the inverse matrix of A and it is denoted by.. Our matrix and vector calculator is the most sophisticated and comprehensive matrix calculator online. You can add, subtract, multiply and transpose matrices. Easily perform matrix multiplication with.. Access to this page has been denied because we believe you are using automation tools to browse the website.

Matrix, ITA's original airfare shopping engine, has yielded years of traveler insights and been the origin for many of our innovative flight shopping features Previous matrix calculators: Determinant of a matrix, Matrix Transpose, Matrix Multiplication This calculator finds modular inverse of a matrix using adjugate matrix and modular multiplicative inverse

This terminology and these facts are very important for matrices too. If you are given a matrix equation like ABÂ = C, where AÂ and C is given and you are to find outÂ B, you would like to “divide off” the matrix A. But we just learned that division is not possible with matrices.$$A =\begin{bmatrix} 1 & 2/3 & -2/3\\ 0 Â & -1/3 & 1/3\\ 2 & 1/3 & -4/3 \end{bmatrix} Â \begin{bmatrix} 1 & 0 & 0\\ 0 Â & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$$ Definition of INVERTIBLE MATRIX in the Definitions.net dictionary. Information and translations of INVERTIBLE MATRIX in the most comprehensive dictionary definitions resource on the web A matrix is invertible if and only if it can be written as the product of elementary reduction matrices. First, elementary matrices are invertible and their inverses are also elementary First of all, to have an inverse the matrix must be "square" (same number of rows and columns).