Podcast
Questions and Answers
What is the rank of the matrix
[4 2 3]
[8 5 2]
[12 -4 5]?
What is the rank of the matrix [4 2 3] [8 5 2] [12 -4 5]?
What value of k makes the rank of the matrix
[1 2 3]
[2 k 7]
[3 6 10] equal to 2?
What value of k makes the rank of the matrix [1 2 3] [2 k 7] [3 6 10] equal to 2?
Using the Gauss Jordan method, which of the following matrices can be inverted?
[ 1 -1 0]
[2 3 4]
[0 3 2]?
Using the Gauss Jordan method, which of the following matrices can be inverted? [ 1 -1 0] [2 3 4] [0 3 2]?
What is the current I₁ in the circuit given the equations: 5I₁ + 10I₃ = 220?
Assuming I₃ = 10A.
What is the current I₁ in the circuit given the equations: 5I₁ + 10I₃ = 220? Assuming I₃ = 10A.
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In a series circuit with two parallel resistors of 4Ω and 3Ω connected to a 6V battery, what is the total resistance?
In a series circuit with two parallel resistors of 4Ω and 3Ω connected to a 6V battery, what is the total resistance?
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Which matrix has a rank that can be determined using the normal form method among the given options?
[1 2 3 2]
[2 4 3 0]
[3 2 1 3]
[6 8 7 5]?
Which matrix has a rank that can be determined using the normal form method among the given options? [1 2 3 2] [2 4 3 0] [3 2 1 3] [6 8 7 5]?
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What is the missing current x₄ in the circuit if the flow into node B is: 5I₁ + 10I₃ = 220?
What is the missing current x₄ in the circuit if the flow into node B is: 5I₁ + 10I₃ = 220?
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Study Notes
Matrices Rank and Inverse
-
Rank Determination: Calculate the rank of matrices using row reduction or echelon forms.
-
Matrix (a):
[ \begin{bmatrix} 4 & 2 & 3 \ 8 & 5 & 2 \ 12 & -4 & 5 \end{bmatrix} ] -
Matrix (b):
[ \begin{bmatrix} 2 & -1 & 0 & 5 \ 0 & 3 & 1 & 4 \end{bmatrix} ] -
Matrix (c):
[ \begin{bmatrix} 2 & 1 & -3 \ 1 & -3 & 1 \end{bmatrix} ] -
Matrix (d):
[ \begin{bmatrix} 1 & 2 & 3 & 0 \ 2 & 4 & 3 & 2 \ 3 & 2 & 1 & 3 \ 6 & 8 & 7 & 5 \end{bmatrix} ] -
Gauss-Jordan Inversion Method: Used to find the inverse of matrices by transforming them into reduced row echelon form.
-
Matrix (a):
[ \begin{bmatrix} 1 & -1 & 0 \ 2 & 1 & 4 \ 0 & 3 & 2 \end{bmatrix} ] -
Matrix (b):
[ \begin{bmatrix} 1 & 1 & 3 & -3 \ 1 & -2 & -4 & -4 \end{bmatrix} ] -
Matrix (c):
[ \begin{bmatrix} 2 & -3 & 5 \ 1 & 1 & 2 \ 1 & 2 & -4 \end{bmatrix} ]
Determining Value of k
- Find ( k ) such that the rank of
[ \begin{bmatrix} 1 & 2 & 3 \ 2 & k & 7 \ 3 & 6 & 10 \end{bmatrix} ] is 2.
Normal Form Rank Calculation
- Use normal form to find the ranks of matrices.
- Matrix (a):
[ \begin{bmatrix} 1 & 2 & 3 & 2 \ 2 & 4 & 3 & 0 \ 3 & 2 & 1 & 3 \ 6 & 8 & 7 & 5 \end{bmatrix} ] - Matrix (b):
[ \begin{bmatrix} 2 & -4 & 3 & -4 & 2 \ 0 & 1 & -1 & 3 & 1 \ 4 & -7 & 4 & -4 & 5 \end{bmatrix} ] - Matrix (c):
[ \begin{bmatrix} 0 & 0 & 1 & -3 & -1 \ 3 & 1 & 1 & 0 & 2 \ 1 & 1 & -2 & 0 & 0 \end{bmatrix} ]
Traffic Network Flow
- Analyze a directed graph with vertices A, B, C, D, and respective flow values.
- Flow connections:
- A → D: 80
- A → B: ( x_1 )
- A → C: ( x_2 )
- B → C: 50
- B → D: ( x_3 )
- C → D: 130
- D → B: ( x_4 )
- D → C: ( x_5 )
- All connections direct towards D, except A → C.
Kirchhoff's Laws for Currents
- Apply Kirchhoff's current and voltage laws to find current values in a circuit.
- Given equations:
- ( 5I₁ + 10I₃ = 220 )
- ( 20I₂ + 10I₃ = 240 )
- ( I₁ + I₂ - I₃ = 0 )
Current Calculation in Series Circuit
- Circuit consists of two parallel resistors connected to a 6V battery:
- Top resistor: 4Ω
- Bottom resistor: 3Ω
- Analyze current distribution to calculate the current in each branch.
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Description
Test your knowledge of matrices with this extensive question bank. This quiz covers finding the rank of matrices and calculating inverses using the Gauss Jordan method. Perfect for students preparing for exams in linear algebra.
