Solving
Could the given matrix be the transition matrix of a regular Markov chain?
\[
\left[\begin{array}{rr}
0.1 & 0.9 \\
1 & 0
\end{array}\right]
\]
Simplifying Matrices
Realize a operação sobre linhas $R_{3} \leftrightarrow R_{1}$ na seguinte matriz:
\[
\left[\begin{array}{cccc}
2 & -6 & -6 & 9 \\
1 & 3 & 9 & -4 \\
-4 & 2 & 0 & 0
\end{array}\right]
\]
Finding the Variables
Find formulas for $X, Y$, and $Z$ in terms of $A, B$, and $C$. It may be necessary to make assumptions about the size of a matrix in order to produce a formula. [Hint Compute the product on the left, and set it equal to the right side.]
\[
\left[\begin{array}{cc}
A & B \\
C & 0
\end{array}\right]\left[\begin{array}{cc}
I & 0 \\
X & Y
\end{array}\right]=\left[\begin{array}{ll}
0 & I \\
Z & 0
\end{array}\right]
\]
Find the formulas for $X, Y$, and $Z$. Note that I represents the identity matrix and 0 represents the zero matrix.
\[
\begin{array}{l}
X= \\
Y= \\
Z=
\end{array}
\]
Solving the System of Equations Using an Inverse Matrix
Use the input-output matrix $A$ and the consumer demand matrix $D$ to solve the matrix equation $(I-A) X=D$ for the total output matrix $X$. (Round your answers to three decimal places).
\[
\begin{array}{c}
A=\left[\begin{array}{ll}
0.4 & 0.2 \\
0.3 & 0.1
\end{array}\right] \text { and } D=\left[\begin{array}{l}
14 \\
16
\end{array}\right] . \\
X=[
\end{array}
\]
Multiplication by a Scalar
What is the result of multiplying the matrix \( A = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \) by the scalar \( k = 4 \)?
Multiplication
Perform the operation $A^{2}$ with the given matrices.
\[
A=\left[\begin{array}{cc}
-6 & 3 \\
-1 & -8
\end{array}\right]
\]
Subtraction
Given two matrices A and B. A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 6 & 7 \end{bmatrix} and B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}. Calculate A - B.
Finding the Determinant of the Resulting Matrix
Find the determinant of the following matrix: \[ A = \begin{bmatrix} 4 & 3 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 4 \end{bmatrix} \]
Finding the Inverse of the Resulting Matrix
What is the sum of the entries in the second column of the inverse of the following matrix? HINT: You don't need to find the entire inverse to answer this question.
\[
\left[\begin{array}{cccc}
1 & -1 & -5 & -1 \\
1 & 1 & 1 & 1 \\
-4 & -1 & 1 & 3 \\
2 & -3 & -1 & -1
\end{array}\right]
\]
Finding the Identity Matrix
Given a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), find the identity matrix \( I \) such that \( AI = IA = A \).
Finding the Scalar multiplied by the Identity Matrix
What is the result of the scalar 5 multiplied by the 2x2 identity matrix?
Addition
Given two matrices \(A = \begin{pmatrix} 2 & 3 \\ 4 & 5 \\ 6 & 7 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}\), find the sum \(A + B\).
Simplifying the Matrix Operation
Given two matrices A = [[2, 3], [4, 5]] and B = [[1, 2], [3, 4]], find the result of 2A - B.
Finding the Determinant of a 2x2 Matrix
Find the determinant of the 2x2 matrix \( A = \begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix} \).
Finding the Determinant of a 3x3 Matrix
1 - Calcule estes determinantes.
a) $\left|\begin{array}{ccc}1 & 2 & 30 \\ 0 & -3 & 4 \\ 5 & 1 & -1\end{array}\right|$
b) $\left|\begin{array}{ccc}7 & 0 & -2 \\ 1 & 3 & 2 \\ 0 & -1 & 1\end{array}\right|$
c) $\left|\begin{array}{ccc}7 & 6 & 5 \\ 2 & 3 & 19 \\ -4 & 11 & 5\end{array}\right|$
d) $\left|\begin{array}{ccc}4 & 5 & 11 \\ 0 & -2 & 17 \\ 0 & 0 & 3\end{array}\right|$
e) $\left|\begin{array}{ccc}0 & 0 & 0 \\ 2 & 3 & 19 \\ -4 & 11 & 5\end{array}\right|$
f) $\left|\begin{array}{ccc}3 & 0 & 0 \\ -5 & 1 & 0 \\ 11 & 6 & 14\end{array}\right|$
Finding the Determinant of Large Matrices
Find the determinant of the following 4x4 matrix: \[A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \\ 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \end{pmatrix}\]
Inverse of a 2x2 Matrix
Find the inverse, if it exists, for the given matrix.
\[
\left[\begin{array}{ll}
-3 & -1 \\
-2 & -1
\end{array}\right]
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The inverse matrix is . (Type a matrix, using an integer or simplified fraction for each matrix element. Do not factor out a scalar multiple.)
B. There is no inverse of the given matrix.
Inverse of an nxn Matrix
Find the inverse of the matrix A = \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \)
Finding Reduced Row Echelon Form
Question 3
25 pts
The matrix
\[
\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 2 \\
0 & 0 & 0
\end{array}\right]
\]
is in reduced form.
True
False
Finding the Transpose
Find the transpose of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \).
Finding the Adjoint
Find the adjoint of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \)
Finding the Cofactor Matrix
2 - Calcule o menor complementar do elemento a23 da matriz.
\[
A=\left[\begin{array}{cccc}
0 & 4 & -2 & 4 \\
-6 & 2 & 10 & 0 \\
5 & 8 & -5 & 2 \\
0 & -2 & 1 & 0
\end{array}\right]
\]
Null Space
Given a matrix A = \(\begin{bmatrix} 2 & 4 & 1 \ 1 & 2 & 1 \ 3 & 6 & 2 \end{bmatrix}\), find the null space.
Finding the Pivot Positions and Pivot Columns
Given the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$, find the pivot positions and pivot columns.
Matrices Addition
Given two matrices: A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. Find the result of A + B.
Matrices Subtraction
Given two matrices A = \(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and B = \(\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}\), find the result of A - B.
Matrices Multiplication
Given the matrices \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \), find the product of \( AB \).
Finding the Trace
Find the trace of the following matrix: \[\begin{pmatrix} 4 & 7 & 3 \ 2 & -1 & 5 \ 1 & 0 & 6 \end{pmatrix}\]
Finding the Basis
Find the basis for the span of the set of vectors \( \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 5 \\ 2 \\ 1 \end{bmatrix} \right\} \).
Finding the Basis and Dimension for the Row Space of the Matrix
Find the basis and dimension for the row space of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 & 4 \ \ 2 & 4 & 6 & 8 \ \ 3 & 6 & 9 & 12 \ \ 4 & 8 & 12 & 16 \end{bmatrix} \)
Finding the LU Decomposition of a Matrix
Find the LU Decomposition of the matrix \(A\) where \(A = \begin{bmatrix} 4 & 3 \ 6 & 3 \end{bmatrix}\)
Diagonalizing a Matrix
Diagonalize the following matrix: \[ A = \begin{pmatrix} 2 & 1 \newline 1 & 2 \end{pmatrix} \]