Matrices: Understanding Systems of Equations with Examples

Find $5 \times 5$ invertible matrix A over $F_{3}$ such that $A^{- 1} = 2 A^{3} + 2 I$ , $A \neq I$

niouzesto 2022-09-11

Consider a nilpotent matrix A of index of nilpotency n , then $A^{n} = O$ and $A^{n - 1} \neq O$ where O denotes null matrix of same order as of A
Now $A^{n} = O ⟹ det (A)^{m} = 0 ⟹ det (A) = 0$
but $A^{n - 1} \neq O ⟹ det (A) \neq 0$
I know I am wrong in the second statement because a matrix not being equal to a null matrix doesn't imply that it has to be necessarily non-singular.
Therefore aren't we allowed to take determinant on both sides for the inequation $A \neq O$ ?

reinzogoq 2022-09-10

I have a set of tuples $S = {(i_{1}, j_{1}, k_{1}), (i_{2}, j_{2}, k_{2}), \dots, (i_{n}, j_{n}, k_{n})}$ . How can I define the matrix M in a formal way:
$M_{i, j} = {\begin{cases} k^{2} + i + j & (i, j, k) \in S \\ - 1 & o t h e r w i s e \end{cases}$
How can I define the matrix in the following two cases:
Case 1: There is no duplicate (i, j) pairs in the set S.
Case 2: There might be some duplicate (i, j) pairs, and I want to choose the pair that has the largest k. For example: S ={(1,1,2), (1,1,5)}, and I want $M_{1, 1} = 5^{2} + 1 + 1$

tragikovas 2022-09-10

I am trying to prove that $A \times (B + C) = A \times B + A \times C .$
I have managed to prove this by finding the LHS
$(\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}) \times (\begin{matrix} b_{1} + c_{1} \\ b_{2} + c_{2} \\ b_{3} + c_{3} \end{matrix})$
then finding the RHS
$(\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}) \times (\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}) + (\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}) \times (\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix})$
and equating LHS = RHS.
However I need to prove the equation directly from the LHS rather than finding the RHS and equating (possibly using separate matrix properties). I.E. I need to continue proving from
$(\begin{matrix} a_{2} b_{3} + a_{2} c_{3} - a_{3} b_{2} - a_{3} c_{2} \\ a_{3} b_{1} + a_{3} c_{1} - a_{1} b_{3} - a_{1} c_{3} \\ a_{1} b_{2} + a_{1} c_{2} - a_{2} b_{1} - a_{2} c_{1} \end{matrix})$

ohgodamnitw0 2022-09-09

Let A be a $n \times n$ square matrix, and
$‖ A ‖_{1} := {max}_{j = 1}^{n} \sum_{i = 1}^{n} | A_{i, j} |$
and
$‖ A ‖_{max} := max_{i, j} | A_{i, j} |$
1)Does $‖ A ‖_{1} \leq 1$ imply that $‖ A ‖_{max} \leq 1$ ?
2)Is there any relation between $‖ A ‖_{1}$ and $‖ A ‖_{max}$ ?

Milton Anderson 2022-09-09

Use the relation $| A B | = | A | | B |$ to show that
$(a_{1}^{2} + a_{2}^{2}) (b_{1}^{2} + b_{2}^{2}) = (a_{1} b_{1} - a_{2} b_{2})^{2} + (a_{2} b_{1} + a_{1} b_{2})^{2} .$

Truscelli3h 2022-09-09

Let $n \in N$ be a (non-zero) natural number and let $d_{1}, \dots, d_{n} > 0$ positive real numbers. Assume that
$\sum_{i = 1}^{n} d_{i} = \sum_{i = 1}^{n} d_{i}^{- 1} = n .$
Does it follow that $d_{i} = 1$ for all $i = 1, 2, \dots, n ?$

cuuhorre76 2022-09-07

Given: $X^{μ v} = (\begin{array}{cccc} 2 & 0 & 1 & - 1 \\ - 1 & 0 & 3 & 2 \\ - 1 & 1 & 0 & 0 \\ - 2 & 1 & 1 & - 2 \end{array})$
Find the component $X_{ν}^{μ}$ so for my understanding it should be just
$X_{ν}^{μ} = X^{μ ρ} X_{ρ ν} = δ_{ν}^{μ}$

Pranav Ward 2022-09-07

Given the Boolean matrices $A_{m \times k}$ , $B_{m \times k}$ and $C_{k \times n}$ , is it true that $(A \oplus B) * C = (A * C) \oplus (B * C)$ , where $\oplus$ denotes elementwise XOR and $*$ denotes Boolean matrix multiplication?
I'm not sure if this make any difference, but the matrices A and B in my case are exponentiations of a graph adjacency matrix , so it will never happen that Bi,j=1 and Ai,j=0.

timberwuf8r 2022-09-06

For $A, B, C, D \in F^{n \times n}$ , show that $det (\begin{matrix} A & B \\ C & D \end{matrix}) = det A det D - det B det C$

Pavukol 2022-09-06

Decide whether A and B are inverses of each other.
$A = \begin{matrix} 3 & - 7 \\ - 2 & 5 \end{matrix}$
$B = \begin{matrix} 5 & 7 \\ 2 & 3 \end{matrix}$
Are A and B inverses of each other yes or no?

potrefilizx 2022-09-05

Show that the set of all 3 by 3 upper triangular matrices forms afinite dimensional subspace of the space of all 3 by 3 real matrices. Determine the dimension and find a basis for the space.

vballa15ei 2022-09-05

Find two matrices A, B such that $A^{2} + B^{2} = [\begin{matrix} 2 & 3 \\ 3 & 2 \end{matrix}] .$

cuuhorre76 2022-09-04

I'm trying to show for $n \times n$ complex matrices A and B that
$s (A B) \geq s (A) s (B)$
where s(M) is the smallest singular value of M.
I've proved this when one of A, B is singular (in this case s(A)s(B)=0).
I've also proved when one of A, B is unitary. Since if B is unitary then s(B)=1 and
$B^{} A^{} A B = s (A B)^{2} v ⟹ A^{} A (B v) = s (A B)^{2} B v$
so $s (A B) \geq s (A) = s (A) s (B)$ . Using $s (M^{}) = s (M)$ , a similar argument goes when A is unitary.

Makayla Reilly 2022-09-04

a) Prove that if A is an $n \times n$ matrix such that $A^{T} = - A$ , then $det (A) = (- 1)^{n} det (A)$
b) Use the result of part (a) to prove that if n is odd, then A is singular.

batystowy2b 2022-09-04

Is there any special reason why $[\begin{matrix} 7 & 2 & 1 \\ - 9 & 0 & 2 \\ 8 & 2 & 1 \end{matrix}]$ and $[\begin{matrix} 0 & 2 & 1 \\ 0 & 0 & 2 \\ 1 & 2 & 1 \end{matrix}]$ have the same determinant of 4?

phoreeldoefk 2022-09-02

Find a matrix X that satisfies the equation BX+AB=CX

badlife18va 2022-08-19

Let $A = (\begin{matrix} \cos 60^{\circ} & \sin 60^{\circ} \\ \sin 60^{\circ} & - \cos 60^{\circ} \end{matrix})$
A can be written like:
$A = (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}) (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (\begin{matrix} \cos (- θ) & - \sin (- θ) \\ \sin (- θ) & \cos (- θ) \end{matrix})$
using a specific $θ$ . Find this $θ$

Kaydence Villegas 2022-08-19

What are the trivial and non-trivial solution of homogeneous equations in the matrices?

Jazmin Clark 2022-08-10

Let $f = (\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 6 & 9 & 5 & 1 & 3 & 2 & 7 & 8 & 4 \end{matrix}) and g = (\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 9 & 7 & 4 & 3 & 8 & 1 & 5 & 2 & 6 \end{matrix})$ be permutations in $S_{9}$ , written in two-line notation. What is fin cycle notation? Enter single spaces between the numbers in each cycle. Do not type spaces anywhere else in your answer.
Let $h = f \cdot g^{- 1}$ . What is the second line of h in two-line notation? Enter it as a list of numbers separated by single spaces.

Even if you have never worked with matrices equations before, it is high time to dive into this interesting topic as you look up various questions and receive professional help by turning to our matrices examples. Remember that these are used in encryption, economics, video games, animation, and all kinds of physics where quantum mechanics and optics are involved. Since there are matrices problems based on equations, we also provide you with the answers that will help you find the most efficient solutions and challenge yourself with more than one mathematical analysis method.

Precalculus

Learn Matrices: Practice Problems and Examples