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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.

55954 questions
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How to use powers on matrices

In the questions compute $\begin{pmatrix} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{pmatrix}^6$ and $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^{99}$, how would you solve these?
user406996
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Least rank of a matrix

Let $A$ be a $7\times 7$ matrix such that it has rank $3$ and $a$ be $7\times 1$ column vector. Then least possible rank of $A+(a a^T)$ is? ($a^T$ is transpose of the vector) Intuitively I think it's $2$ as $a a^T$ has rank $1$, and the most it…
Nitish
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Matrix norm of two hermitian matrices.

Let A and B be two hermitian matrices. Let $|||\cdot|||$ be any induced matrix norm. I hope to find some upper bound inequalities or relationship of the matrix norm: $|||iA - B|||$. The only thing I know is that it is $|||iA - B|||\leq |||A||| +…
Ka Wa Yip
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Matrix and Equations::

The cost of $3$ copies and $5$ pens is Rs. $21$. The cost of $2$ copies and $3$ pens is Rs. $13$. Find the cost of each copy and pen using matrix method. My Attempt: Let the cost of each copy and each pen be $x$ and $y$…
pi-π
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Find all matrices $X$ such that $ABXB^tA^t=I$

Find all matrices $X$ such that: $$ABXB^tA^t=I$$ if $A=\begin{pmatrix} 1 &-2 &2\\ 3 &-5 &6\\ -1 &2 &-1 \end{pmatrix}$ and $B=\begin{pmatrix} -3 &-2 &-2\\ 2 & 1 &1\\ 6 &3 &4 \end{pmatrix}$. So I managed to get that $AB=\begin{pmatrix} 5 &2 &4\\ 17…
lmc
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Function of this matrix

Given a matrix M=$ \begin{bmatrix} 2 & 1 \\ 1 & 2 \\ \end{bmatrix}$ How do I find $ \cos ({Mπ \over 6})$
Pruthvi Bharadwaj
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Equal nullspaces of consecutive powers of a matrix

The question states: "Consider a square matrix $A$ with $\ker(A^2) = \ker(A^3)$. Is $\ker(A^3)= \ker(A^4)$? Justify your answer." It's clear that if $x$ satisfies $A^3x = 0$, then it satisfies $A^4x = 0$. I am wondering whether the opposite is true.
Squirrel-Power
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Does the fact that $A^{17} = I_2$ imply that the matrix $A$ must be $I_2$?

Does the fact that $A^{17} = I_2$ imply that the matrix $A$ must be $I_2$? Since the question does not specify whether the entries of $A$ are allowed to be complex valued functions, I said that IF the entries can indeed be complex numbers, then the…
Squirrel-Power
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Absolute value of a matrix

Let $A=(a_{ij})$ be an infinite matrix. Consider $|A|=(A^*A)^{1/2}$ and $A'=(|a_{ij}|)$. Is there any relation between $|A|$ and $A'$?
niki
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Square roots of a $3\times3$ matrix

Here is $3\times3$ matrix$$\begin{pmatrix} 1& -18& 0\\ 0 & 4& 0\\ -8& -13 & 9\end{pmatrix}$$ How can I find two different matrices so that $R^2=A$?
Ioanah
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Rotation matrix

HI I am wondering if there is a unique matrix that maps $(x_1,y_1,z_1)$ into $(x_2,y_2,z_2)$. These two vectors have equal magnitude and are defined in orthogonal 3-D basis. If there is a unique solution how can I find it by considering rotation…
Ankit
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A question about Hadamard matrices

Is it possible to find a matrix $A$ such that: $$\exp(A)+\exp(A^{-1})=H_2$$ with $A$ a $2\times 2$ matrix and $H_2$ a Hadamard matrix? The result can be extended to every Hadamard matrix $H_N$ with $N$ power of two? Thanks in advance.
Riccardo.Alestra
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Proving an equality of matrices

I am asked to prove that $$\begin{pmatrix} \\ A & B\\ C &D\end{pmatrix}^{-1}=\begin{pmatrix} M & -MBD^{-1} \\ -D^{-1}CM & D^{-1}+D^{-1}CMBD^{-1} \end{pmatrix}$$ Where $M=(A-BD^{-1}C)^{-1}$. Unfortunately, I have no idea what to do about it.
Gigili
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Exponential and Q matrice question

For any matrix $Q$, show that $$\det(e^{Q})=e^{\operatorname{tr}Q}$$ where tr represents the trace and det is the determinant
user39325
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What fractions can fill a $N$ by $N$ matrix given that their sum is always 2?

My problem is the following: I have a matrix $N$ by $N$ in size. I want to fill it with fractions of $1$ of increasing denominator in relation to their distance from the center of the matrix. The central value is always $1$. $N$ is always…
KappaG3
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