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Application of Matrices and Determinants - Two Marks Study Materials

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 30
    15 x 2 = 30
  1. If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A−1.

  2. If A is a non-singular matrix of odd order, prove that |adj A| is positive

  3. If A is symmetric, prove that then adj Ais also symmetric.

  4. If adj(A) = \(\left[ \begin{matrix} 0 & -2 & 0 \\ 6 & 2 & -6 \\ -3 & 0 & 6 \end{matrix} \right] \), find A−1.

  5. Find the rank of each of the following matrices:
    \(\left[ \begin{matrix} 3 & 2 & 5 \\ 1 & 1 & 1 \\ 3 & 3 & 6 \end{matrix} \right] \)
     

  6. Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.

  7. Find the rank of the following matrices by minor method:
    \(\left[ \begin{matrix} -1 & 3 \\ 4 & -7 \\ 3 & -4 \end{matrix} \right] \)

  8. Find the rank of the following matrices by minor method:
    \(\left[ \begin{matrix} 1 \\ 3 \end{matrix}\begin{matrix} -2 \\ -6 \end{matrix}\begin{matrix} -1 \\ -3 \end{matrix}\begin{matrix} 0 \\ 1 \end{matrix} \right] \)

  9. Find the rank of the following matrices which are in row-echelon form :
    \(\left[ \begin{matrix} 6 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} 0 \\ \begin{matrix} 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} -9 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \right] \)

  10. For any 2 x 2 matrix, if A (adj A) =\(\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right] \) then find |A|.

  11. For the matrix A, if A3 = I, then find A-1.

  12. If A is a square matrix such that A3 = I, then prove that A is non-singular.

  13. Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).

  14. Solve : 2x - y = 3, 5x + y = 4 using matrices.

  15. Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

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