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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.

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$3\times3$ matrix inverse and multiplication

Which answer is correct? Mine or this from the textbook?
Adam Rubinson
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How to do this line rotation (matrices) question?

Question: The line $y=10-2x$ is rotated anticlockwise about the origin such that its image has an $x$-intercept of $(p,0)$ and $y$-intercept of $(0,q)$. Determine the angle of rotation θ $(0 ≤ θ ≤ 90)$ such that $p=q$. I know that i probably need…
CountDOOKU
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Let $A$ and $B$ be $n \times n$ real matrices such that det$(A) > 0$ and det$(B) < 0$. For $0≤ t ≤ 1$ let $C(t) = tA + (1-t)B$

Let $A$ and $B$ be $n \times n$ real matrices such that det$(A) > 0$ and det$(B) < 0$. For $0≤ t ≤ 1$ let $C(t) = tA + (1-t)B$. Then there exists exactly one $t_{0}$ in $(0,1)$ such that $C(t_{0})$ is not invertible. (True/false) I took two matrices…
Mathaddict
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Properties of Pauli Matrices

Using the three Pauli spin matrices $$\boldsymbol{\sigma}_{1}=\left(\begin{array}{ll}{0} & {1} \\ {1} & {0}\end{array}\right), \quad \boldsymbol{\sigma}_{2}=\left(\begin{array}{cc}{0} & {-i} \\ {i} & {0}\end{array}\right), \quad…
rami_salazar
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Matrix rearrangement

I have a matrix A in this form: $$ A= \left[ \begin{array}{cccc} x_1 & x_2 & 0 & 0 \\ 0 & 0& x_1 & x_2 \end{array} \right] $$ Here, $x_1$ and $x_2$ are variables, and A is a $2 \times 4$ matrix. I would like to rearrange the matrix product so…
FnitDuan
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When $e$ is an eigenvector to $G$ prove that $e$ is an eigenvector of $G+k I$ and $G^2$

I've just started learning matrices and I've been shown how to perform row operations, how to find an inverse matrix, how to find eigenvectors from a given matrix, reduced-echelon form, basis, dim(M), rank(M), etc. However, I've just come across…
Ozzy
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Why does a non square matrix lack a multiplicative identity

Precisely, why is multiplicative identity defined to be $IA=AI=A$ why both sides should work why not something like $AI=A$ ? Is there an underlying advantage?
Aditya Prakash
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Matrix, satisfying $A^T=p(A)$

Let $A$ be real square matrix, satisfying $A^T=p(A)$ for some polynomial $$p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$ such that $a_0\neq 0$. I have to prove that $A$ is invertible and I have no idea how to do it.
Hrackadont
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Find the normal vector to the projection plane

In the context of perspective projection. Given focal length is $2.387$, the camera is at $(0.0.0)$ looking at $-z$ direction A rectangle lies on a plane tilted from view plane. Also given the projected point on the view plane is…
fiftyplus
  • 123
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Uniqueness of the cube root of a symmetric positive definite matrix

After doing many calculus steps I found that $S^3=2I$ where $S\in \mathcal{M}_{m,m}$ is a symmetric positive definite matrix. So I'm wondering if I can write my solution $S\propto I$.
user2987
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Number of ones in Binary matrix multiplication

Consider a binary matrix $\mathbf A_n$ corresponding to values $0$ to $2^n-1$ where each row represents a length $n$ binary representation of a real number. For example, for $n=3$ we have $\mathbf A_3=\begin{bmatrix} 0 & 0 & 0\\ 1 …
nOp
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The number of permutation matrices of order $n$

The number of permutation matrices of order $n$, where permutation matrices are given by $$A=[a_{i,j}],~a_{i,j}=0~\mbox{or}~1,~\mbox{so that}~\sum_{i=1}^na_{i,j}=\sum_{j=1}^na_{i,j}=1,~\forall i,j=1,2,...,n$$ Can it be $n!$ ?
Riaz
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Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be an $n \times n$ identity matrix. Prove that the matrix $I −A$ is an idempotent matrix

A square matrix $A$ is called idempotent if $A^2 = A$. (a) Suppose $A$ is an $n × n$ idempotent matrix and let $I$ be the $n × n$ identity matrix. Prove that the matrix $I −A$ is an idempotent matrix. (b) Assume that $A$ is an $n×n$ non zero…
john Doe
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Cramer's Rule with Row Reduction - Getting Different Determinants

I've got to find the y value using Cramer's rule from the following set of equations: $$ 4x - 2y - 3z = 5 $$ $$ -2x - 4y + z = 21 $$ $$ 8x - y - 2z = 7 $$ I'm stuck on the part that requires finding the determinant of the top matrix (which should be…
Kyzen
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Is it possible to have a $2\times2$ matrix such that $A^3=0\ne A^2$?

I know it is possible to have such a matrix if $A$ is $3\times 3$. I think it won't be possible in this case but I'm not sure how to prove a general case. A specific case proof would work something like this. $A$ is determined by its action on…
Anvit
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