Exam Questions & Step-by-Step Markschemes (4 Problems)
Question 1 • 2024
4 Marks
If A = \begin{pmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{pmatrix} and B = \begin{pmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{pmatrix} , find the product AB . Hence, solve the system of linear equations: \begin{aligned} x - y + 2z &= 1 \\ 2y - 3z &= 1 \\ 3x - 2y + 4z &= 2 \end{aligned}
Answer: x = 0, \quad y = 5, \quad z = 3
Question 2 • 2024
4 Marks
Express the matrix A = \begin{pmatrix} 3 & 5 \\ 1 & -1 \end{pmatrix} as the sum of a symmetric matrix and a skew-symmetric matrix.
Answer: A = \begin{pmatrix} 3 & 3 \\ 3 & -1 \end{pmatrix} + \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}
Question 3 • 2023
4 Marks
જો શ્રેણિક A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} હોય, તો સાબિત કરો કે A^2 - 6A + 11I = O અને તે પરથી A^{-1} મેળવો.
Answer: A^{-1} = \frac{1}{11} \begin{bmatrix} 4 & 1 \\ -3 & 2 \end{bmatrix}
Question 4 • 2024
4 Marks
Find the inverse of the matrix A = \begin{pmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{pmatrix} and hence solve the system of equations: \begin{aligned} x - y + 2z &= 1 \\ 2y - 3z &= 1 \\ 3x - 2y + 4z &= 2 \end{aligned}
Answer: x = 0, \, y = 5, \, z = 3