Matrices Question Bank

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.