Matrices

Solving

Could the given matrix be the transition matrix of a regular Markov chain?

[\begin{array}{rr} 0.1 & 0.9 \\ 1 & 0 \end{array}]

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Simplifying Matrices

Realize a operação sobre linhas

R_{3} \leftrightarrow R_{1}

na seguinte matriz:

[\begin{array}{cccc} 2 & - 6 & - 6 & 9 \\ 1 & 3 & 9 & - 4 \\ - 4 & 2 & 0 & 0 \end{array}]

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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.]

[\begin{array}{cc} A & B \\ C & 0 \end{array}] [\begin{array}{cc} I & 0 \\ X & Y \end{array}] = [\begin{array}{ll} 0 & I \\ Z & 0 \end{array}]

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}

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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{matrix} A = [\begin{array}{ll} 0.4 & 0.2 \\ 0.3 & 0.1 \end{array}] and D = [\begin{array}{l} 14 \\ 16 \end{array}] . \\ X = [ \end{matrix}

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Multiplication by a Scalar

What is the result of multiplying the matrix

A = [\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix}]

by the scalar

k = 4

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Multiplication

Perform the operation

A^{2}

with the given matrices.

A = [\begin{array}{cc} - 6 & 3 \\ - 1 & - 8 \end{array}]

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Subtraction

Given two matrices A and B. A =

[\begin{matrix} 2 & 3 \\ 4 & 5 \\ 6 & 7 \end{matrix}]

and B =

[\begin{matrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{matrix}]

. Calculate A - B.

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Finding the Determinant of the Resulting Matrix

Find the determinant of the following matrix:

A = [\begin{matrix} 4 & 3 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 4 \end{matrix}]

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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.

[\begin{array}{cccc} 1 & - 1 & - 5 & - 1 \\ 1 & 1 & 1 & 1 \\ - 4 & - 1 & 1 & 3 \\ 2 & - 3 & - 1 & - 1 \end{array}]

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Finding the Identity Matrix

Given a 2x2 matrix

A = [\begin{matrix} a & b \\ c & d \end{matrix}]

, find the identity matrix

I

such that

A I = I A = A

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Finding the Scalar multiplied by the Identity Matrix

What is the result of the scalar 5 multiplied by the 2x2 identity matrix?

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Addition

Given two matrices

A = (\begin{matrix} 2 & 3 \\ 4 & 5 \\ 6 & 7 \end{matrix})

and

B = (\begin{matrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{matrix})

, find the sum

A + B

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Simplifying the Matrix Operation

Given two matrices A = [[2, 3], [4, 5]] and B = [[1, 2], [3, 4]], find the result of 2A - B.

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Finding the Determinant of a 2x2 Matrix

Find the determinant of the 2x2 matrix

A = [\begin{matrix} 3 & 4 \\ 2 & 5 \end{matrix}]

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Finding the Determinant of a 3x3 Matrix

1 - Calcule estes determinantes. a)

| \begin{array}{ccc} 1 & 2 & 30 \\ 0 & - 3 & 4 \\ 5 & 1 & - 1 \end{array} |

| \begin{array}{ccc} 7 & 0 & - 2 \\ 1 & 3 & 2 \\ 0 & - 1 & 1 \end{array} |

| \begin{array}{ccc} 7 & 6 & 5 \\ 2 & 3 & 19 \\ - 4 & 11 & 5 \end{array} |

| \begin{array}{ccc} 4 & 5 & 11 \\ 0 & - 2 & 17 \\ 0 & 0 & 3 \end{array} |

| \begin{array}{ccc} 0 & 0 & 0 \\ 2 & 3 & 19 \\ - 4 & 11 & 5 \end{array} |

| \begin{array}{ccc} 3 & 0 & 0 \\ - 5 & 1 & 0 \\ 11 & 6 & 14 \end{array} |

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Finding the Determinant of Large Matrices

Find the determinant of the following 4x4 matrix:

A = (\begin{matrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \\ 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \end{matrix})

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Inverse of a 2x2 Matrix

Find the inverse, if it exists, for the given matrix.

[\begin{array}{ll} - 3 & - 1 \\ - 2 & - 1 \end{array}]

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.

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Inverse of an nxn Matrix

Find the inverse of the matrix A =

[\begin{matrix} 1 & 2 3 & 4 \end{matrix}]

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Finding Reduced Row Echelon Form

Question 3 25 pts The matrix

[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array}]

is in reduced form. True False

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Finding the Transpose

Find the transpose of the matrix

A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]

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Finding the Adjoint

Find the adjoint of the matrix

A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]

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Finding the Cofactor Matrix

2 - Calcule o menor complementar do elemento a23 da matriz.

A = [\begin{array}{cccc} 0 & 4 & - 2 & 4 \\ - 6 & 2 & 10 & 0 \\ 5 & 8 & - 5 & 2 \\ 0 & - 2 & 1 & 0 \end{array}]

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Null Space

Given a matrix A =

[\begin{matrix} 2 & 4 & 1 1 & 2 & 1 3 & 6 & 2 \end{matrix}]

, find the null space.

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Finding the Pivot Positions and Pivot Columns

Given the matrix

A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]

, find the pivot positions and pivot columns.

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Matrices Addition

Given two matrices: A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. Find the result of A + B.

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Matrices Subtraction

Given two matrices A =

(\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix})

and B =

(\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix})

, find the result of A - B.

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Matrices Multiplication

Given the matrices

A = [\begin{matrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{matrix}]

and

B = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}]

, find the product of

A B

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Finding the Trace

Find the trace of the following matrix:

(\begin{matrix} 4 & 7 & 3 2 & - 1 & 5 1 & 0 & 6 \end{matrix})

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Finding the Basis

Find the basis for the span of the set of vectors

{[\begin{matrix} 1 \\ 2 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 3 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 3 \\ 5 \\ 2 \\ 1 \end{matrix}]}

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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{matrix} 1 & 2 & 3 & 4 2 & 4 & 6 & 8 3 & 6 & 9 & 12 4 & 8 & 12 & 16 \end{matrix}]

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Finding the LU Decomposition of a Matrix

Find the LU Decomposition of the matrix

A

where

A = [\begin{matrix} 4 & 3 6 & 3 \end{matrix}]

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Diagonalizing a Matrix

Diagonalize the following matrix:

A = (\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix})

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Solving

Simplifying Matrices

Finding the Variables

Solving the System of Equations Using an Inverse Matrix

Multiplication by a Scalar

Multiplication

Subtraction

Finding the Determinant of the Resulting Matrix

Finding the Inverse of the Resulting Matrix

Finding the Identity Matrix

Finding the Scalar multiplied by the Identity Matrix

Addition

Simplifying the Matrix Operation

Finding the Determinant of a 2x2 Matrix

Finding the Determinant of a 3x3 Matrix

Finding the Determinant of Large Matrices

Inverse of a 2x2 Matrix

Inverse of an nxn Matrix

Finding Reduced Row Echelon Form

Finding the Transpose

Finding the Adjoint

Finding the Cofactor Matrix

Null Space

Finding the Pivot Positions and Pivot Columns

Matrices Addition

Matrices Subtraction

Matrices Multiplication

Finding the Trace

Finding the Basis

Finding the Basis and Dimension for the Row Space of the Matrix

Finding the LU Decomposition of a Matrix

Diagonalizing a Matrix

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