Practice Set-1

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0. Basic Operations

  1. Add the matrices: \[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \]
  2. Subtract the matrices: \[ \begin{bmatrix} 4 & -2 \\ 1 & 0 \end{bmatrix} - \begin{bmatrix} 2 & 3 \\ -1 & 5 \end{bmatrix} \]
  3. Multiply the matrices: \[ \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 2 & 0 \\ 3 & 4 \end{bmatrix} \]
  1. Find the determinant of: \[ \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \]
  2. Evaluate the determinant of: \[ \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix} \]
  1. Find the inverse of: \[ \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix} \]
  2. Show that the matrix is not invertible: \[ \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} \]
  1. Identify if the following is symmetric, skew-symmetric, or neither: \[ \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix} \]
  2. Write a \(3 \times 3\) diagonal matrix and a scalar matrix with scalar \(5\).
  3. Verify if the identity matrix \[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] satisfies \(AI = IA = A\) for \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}. \]

A. Row Reduced Echelon Form (RREF)

  1. Reduce the following matrix to row reduced echelon form: \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix} \]
  2. Find the RREF of: \[ B = \begin{bmatrix} 2 & 1 & -1 \\ -1 & 0 & 3 \\ 4 & 1 & 2 \end{bmatrix} \]
  3. Bring to RREF: \[ C = \begin{bmatrix} 0 & 2 & 4 & 6 \\ 1 & 3 & 5 & 7 \\ 2 & 4 & 6 & 8 \end{bmatrix} \]

B. Rank of a Matrix

  1. Find the rank of: \[ D = \begin{bmatrix} 1 & 0 & -1 & 2 \\ 2 & 1 & 0 & 3 \\ 3 & 1 & -1 & 5 \end{bmatrix} \]
  2. Compute the rank of: \[ E = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
  3. Find the rank of: \[ F = \begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

C. System of Linear Equations

  1. Solve using row-reduction: \[ \begin{cases} x + 2y + 3z = 9 \\ 2x + 3y + z = 8 \\ 3x + y + 2z = 7 \end{cases} \]
  2. Determine if the system has unique, infinite, or no solution: \[ \begin{cases} x + y + z = 6 \\ 2x + 2y + 2z = 12 \\ 3x + 3y + 3z = 18 \end{cases} \]
  3. Solve the system: \[ \begin{cases} x + y - z = 1 \\ 2x - y + z = 3 \\ 3x + 2y = 4 \end{cases} \]
  4. A system of equations is given by: \[ \begin{cases} x_1 + x_2 + x_3 + x_4 = 5 \\ 2x_1 + 3x_2 + 5x_3 + 7x_4 = 15 \\ x_1 - x_2 + 2x_3 + 3x_4 = 4 \end{cases} \]