Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

A matrix is a rectangular array of elements, usually numbers or variables, arranged in rows and columns. A matrix with $m$ rows and $n$ columns has $m \times n$ elements and is called an $m$ by $n$ matrix. Matrices are a part of .

Matrices can be added and subtracted. Furthermore, if they have compatible shapes, they can be multiplied. More precisely, given two matrices $A$ and $B$, the matrix $AB$ is defined when the number of columns of $A$ is equal to the number of rows of $B$. In particular, given a natural number $n$, any two matrices $A$ and $B$ with $n$ columns and $n$ rows can be multiplied in both ways (that is, both $AB$ and $BA$ exist).

For questions specifically concerning matrix equations, use the tag.

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Finding a basis for the intersection of two vector subspaces

Q1. Let $W1=\bigl\{\begin{bmatrix} a & a\\ b & a\end{bmatrix}\mid a,b\in\mathbb{R}\bigr\}$ and $W2=\bigl\{\begin{bmatrix} c&d\\ e& 0\end{bmatrix}\mid c,d,e\in\mathbb{R}\bigr\}$. Q2. Let $W1=\bigl\{\begin{bmatrix} a & a\\ b & b\end{bmatrix}\mid…
Essie
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Evaluation of a certain polynomial on a matrix.

Given $A=\left[\begin{matrix}\dfrac{-1+i\sqrt{3}}{2i}&\dfrac{-1-i\sqrt{3}}{2i}\\\dfrac{1+i\sqrt{3}}{2i}&\dfrac{1-i\sqrt{3}}{2i}\end{matrix}\right]$ where $i=\sqrt{-1}$. Also Given$f(x)=x^2+2.$ Find $f(A)$ My approach: A can be written as…
Saradamani
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how to show these two matrices commutes

could any one tell me how to show $(A-\lambda E)^{-1}E$ and $(A-\lambda E)^{-1}A$ commutes where $\lambda$ is chosen in way such that $(A-\lambda E)$ is invertible? I tried $I=(A-\lambda E)^{-1}(A-\lambda E)=(A-\lambda E)(A-\lambda…
Myshkin
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If A and B are n-th row square matrices, $AB=BA$ and $B^2=0$ prove that $(A+B)^k=A^k+kA^{k-1}B$

If A and B are n-th row square matrices, $AB=BA$ and $B^2=0$ prove that $$(A+B)^k=A^k+kA^{k-1}B$$ Any ideas?
crylikeacanary
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How do I find out the principal minors of a $4\times 4$ matrix?

How many principal minors can a $4\times 4$ matrix have? Is there any general method using which I can found out the principal minors of any $n\times n$ matrix?
WorldGov
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Quick proof of Cauchy-Binet Identity

I am currently preparing a talk for undergraduates during which I will have to resort to the Cauchy-Binet Formula. However, I don't expect most of the audience to be familiar with this identity and, since I would like to make my talk as…
Doom
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example of matrices

Show that $\forall n,k\geq 2$, there are invertible non diagonal matrices $\in M_{n}(\mathbb{R})$ such that $A_{1}^{-1}+A_{2}^{-1}+\cdots +A_{k}^{-1}=(A_{1}+A_{2}+\cdots +A_{k})^{-1}$. I only need one example of $2 \times 2$ matrix with this…
Christian
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Laplace transform table for matrix?

Given a homogeneous linear time invariant dynamical system x' = Ax, by performing Laplace transform, $L[x'(t)] = L[Ax(t)]$ $sX - x(0) = AX$ $sX - AX = x(0)$ $(sI - A)X = x(0)$ $X = (sI-A)^{-1}x(0)$ and by after taking a inverse Laplace…
drerD
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Problem with generalized eigenvectors in a 3x3 matrix.

I have this matrix: $$ A= \begin{pmatrix} 0 & 1 & 1 \\ 0 & 1 & 0 \\ -1 & 1 & 2 \\ \end{pmatrix} $$ I have founded the eigenvalues: $$\lambda_{1,2,3}=1$$ So $$\lambda=1$$$$\mu=3$$ I'm expecting to have one eigenvector plus two…
muserock92
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Finding the rank of the coefficient matrix

Suppose $x = 0$ is the only solution to the matrix equation $Ax = 0$ where $A$ is $m \times n$, $x$ is $n \times 1$, and $0$ is $m \times 1$. Then (i) The rank of $A$ is $n$, and (ii) $m \times n$ Why is this the case, can somebody give me an…
Meera Unni
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To find the number of matrices

Is there any formula to find the number of all possible matrices of order n × n with each entry 0 or 1 ?
Shona
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The derivation of the Binet-Cauchy Formula of Determinants from matrix multiplication

May you explain what happened in the last 3 equalities? The author used Leibniz formula for determinants, right?
Abdu Magdy
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Does exist some formal name for 3 dimentional "matrix"?

What I am looking for is a name in mathematics or even a part of mathematics which has some interpretation for an cubic form of matrices like $2 \times 2 \times 2$ or $3 \times 3 \times 2$ or $3 \times 3 \times 3$ or any like that? For example $2…
koralgooll
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Sum inverse implies equal determinant

Let,$A,B$ are two $n\times n,(n\ge 2)$ non-singular matrices with real entries $(a)$ If $A^{-1}+B^{-1}=(A+B)^{-1}$ , then show that $\text{det(A)}=\text{det(B)}.$ I am getting $$B^{-1}A+A^{-1}B=-I_n.$$ Then how to proceed
debdutt
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For three matrices $A, B$ and $C$ of the same order, if $A$=$B$, then $AC$=$BC$, but converse is not true.

For three matrices $A, B$ and $C$ of the same order, if $A$=$B$, then $AC$=$BC$, but converse is not true. I guess $A,B,C$ all are square matrices of the same order. $$ A=B $$ multiplying by $C$ from right side, $$ \implies AC=BC $$ But why is the…
Sooraj S
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