Determinants

  1. Compute the determinant by cofactor expansion.  solution

    2. \( \begin{vmatrix}\begin{array}{rrrr}{1} & {-2} & {5} & {2} \\ {0} & {0} & {3} & {0} \\ {2} & {-4} & {-3} & {5} \\ {2} & {0} & {3} & {5}\end{array}\end{vmatrix} \)


  2. Compute det(A) where A is given below.    solution

    3. \(A=\left[\begin{array}{rrrr}{2} & {-8} & {6} & {8} \\ {3} & {-9} & {5} & {10} \\ {-3} & {0} & {1} & {-2} \\ {1} & {-4} & {0} & {6}\end{array}\right]\)


  3. Show that if \( \mathrm{det}(A^T) = \mathrm{det}(A^{-1})\) then \(\mathrm{det}(A)=\pm1. \)    proof


  4. Let A and B be \(4 \times 4\) matrices, with  \(\operatorname{det}(A) = -3\) and \(\operatorname{det} (B) = -1 \). Compute:   solution

    5. a. \(\operatorname{det} (AB)\)    b. \(\operatorname{det} (B^{5})\)    c. \(\operatorname{det} (2 A)\)    d. \(\operatorname{det} (A^{T} B A)\)    e. \(\operatorname{det} (B^{-1} A B)\)