Answers for Exercises

Section 1: Determinants

Section 2: Cofactors

Section 3: Cramer's Rule

Section 4: Partitioned Matrices

Section 5: Elementary Matrices

Section 6: Inverse Matrices

1. 1. The determinant of matrix A is -1. As det(A)≠0, matrix A is invertible.
   2. The determinant of matrix B is -1. As det(B)≠0, matrix B is invertible.
   3. The determinant of matrix C is 1. As det(C)$\neq$0, matrix C is invertible.\\
   4. The determinant of matrix D is 0. As det(D)=0, matrix D is not invertible.\\
2. 1. Adjoin the matrix with an identity matrix. \[ \begin{bmatrix} 1 & 2 & -3 & | & 1 & 0 & 0 \\ 0 & 5 & -1 & | & 0 & 1 & 0 \\ -1 & 2 & 2 & | & 0 & 0 & 1 \\ \end{bmatrix} \] Add the row 1 to row 3. Multiply row 2 by 1/5. \[ \begin{bmatrix} 1 & 2 & -3 & | & 1 & 0 & 0 \\ 0 & 1 & -\frac{1}{5} & | & 0 & \frac{1}{5} & 0 \\ 0 & 4 & -1 & | & 1 & 0 & 1 \\ \end{bmatrix} \] Add -4 times row 2 to row 3. \[ \begin{bmatrix} 1 & 2 & -3 & | & 1 & 0 & 0 \\ 0 & 1 & -\frac{1}{5} & | & 0 & \frac{1}{5} & 0 \\ 0 & 0 & -\frac{1}{5} & | & 1 & -\frac{4}{5} & 1 \\ \end{bmatrix} \] Multiply row 3 by -5. \[ \begin{bmatrix} 1 & 2 & -3 & | & 1 & 0 & 0 \\ 0 & 1 & -\frac{1}{5} & | & 0 & \frac{1}{5} & 0 \\ 0 & 0 & 1 & | & -5 & 4 & -5 \\ \end{bmatrix} \] Add 1/5 times row 3 to row 2. Add 3 times row 3 to row 1. \[ \begin{bmatrix} 1 & 2 & 0 & | & -14 & 12 & -15 \\ 0 & 1 & 0 & | & -1 & 1 & -1 \\ 0 & 0 & 1 & | & -5 & 4 & -5 \\ \end{bmatrix} \] Add -2 times row 2 to row 1. \[ \begin{bmatrix} 1 & 0 & 0 & | & -12 & 10 & -13 \\ 0 & 1 & 0 & | & -1 & 1 & -1 \\ 0 & 0 & 1 & | & -5 & 4 & -5 \\ \end{bmatrix}$ \] \[ inv(A)=\begin{bmatrix} -12 & 10 & -13 \\ -1 & 1 & -1 \\ -5 & 4 & -5 \\ \end{bmatrix} \]
   2. Adjoin the matrix with an identity matrix. \[ \begin{bmatrix} -3 & 3 & 2 & | & 1 & 0 & 0\\ -1 & -2 & -1 & | & 0 & 1 & 0\\ 0 & 2 & 1 & | & 0 & 0 & 1\\ \end{bmatrix} \] Multiply row 1 by -1/3. \[ \begin{bmatrix} 1 & -1 & -\frac{2}{3} & | & -\frac{1}{3} & 0 & 0\\ -1 & -2 & -1 & | & 0 & 1 & 0\\ 0 & 2 & 1 & | & 0 & 0 & 1\\ \end{bmatrix} \] Add row 1 to row 2. \[ \begin{bmatrix} 1 & -1 & -\frac{2}{3} & | & -\frac{1}{3} & 0 & 0\\ 0 & -3 & -\frac{5}{3} & | & -\frac{1}{3} & 1 & 0\\ 0 & 2 & 1 & | & 0 & 0 & 1\\ \end{bmatrix} \] Multiply row 2 by -1/3. \[ \begin{bmatrix} 1 & -1 & -\frac{2}{3} & | & -\frac{1}{3} & 0 & 0\\ 0 & 1 & \frac{5}{9} & | & \frac{1}{9} & -\frac{1}{3} & 0\\ 0 & 2 & 1 & | & 0 & 0 & 1\\ \end{bmatrix} \] Add -2 times row 2 to row 3. \[ \begin{bmatrix} 1 & -1 & -\frac{2}{3} & | & -\frac{1}{3} & 0 & 0\\ 0 & 1 & \frac{5}{9} & | & \frac{1}{9} & -\frac{1}{3} & 0\\ 0 & 0 & 1 & | & 2 & -6 & -9\\ \end{bmatrix} \] Multiply row 3 by -9. Add -5/9 times row 3 to row 2. Add 2/3 times row 3 to row 1. \[ \begin{bmatrix} 1 & -1 & 0 & | & 1 & -4 & -6\\ 0 & 1 & 0 & | & -1 & 3 & 5\\ 0 & 0 & 1 & | & 2 & -6 & -9\\ \end{bmatrix} \] Add row 2 to row 3. \[ \begin{bmatrix} 1 & 0 & 0 & | & 0 & -1 & -1\\ 0 & 1 & 0 & | & -1 & 3 & 5\\ 0 & 0 & 1 & | & 2 & -6 & -9\\ \end{bmatrix} \] \[ inv(B)=\begin{bmatrix} 0 & -1 & -1\\ -1 & 3 & 5\\ 2 & -6 & -9 \end{bmatrix} \]
4. 1. det(A)=-1

      cof(A11)=$(-1)^{(i+j)}$det(A11)=$(-1)^{(1+1)}((0*-2)-(1*3))=-3$

      cof(A12)=$(-1)^{(i+j)}$det(A12)=$(-1)^{(1+2)}((-2*-2)-(2*3))=2$

      cof(A13)=$(-1)^{(i+j)}$det(A13)=$(-1)^{(1+3)}((-2*1)-(0*2))=-2$

      cof(A21)=$(-1)^{(i+j)}$det(A21)=$(-1)^{(2+1)}((3*-2)-(1*-1))=5$

      cof(A22)=$(-1)^{(i+j)}$det(A22)=$(-1)^{(2+2)}((3*-2)-(2*-1))=-4$

      cof(A23)=$(-1)^{(i+j)}$det(A23)=$(-1)^{(2+3)}((3*1)-(2*3))=3$

      cof(A31)=$(-1)^{(i+j)}$det(A31)=$(-1)^{(3+1)}((3*1)-(2*3))=9$

      cof(A32)=$(-1)^{(i+j)}$det(A32)=$(-1)^{(3+2)}((3*1)-(2*3))=-7$

      cof(A33)=$(-1)^{(i+j)}$det(A33)=$(-1)^{(3+3)}((3*1)-(2*3))=6$

      \[ inv(A)=\frac{1}{det(A)}$adj(A)=$\frac{1}{det(A)}$ M(A)'=-1 \begin{bmatrix} -3 & 2 & -2 \\ 5 & -4 & 3 \\ 9 & -7 & 6 \\ \end{bmatrix}' =-1\begin{bmatrix} -3 & 5 & 9 \\ 2 & -4 & -7 \\ -2 & 3 & 6 \\ \end{bmatrix} =\begin{bmatrix} 3 & -5 & -9 \\ -2 & 4 & 7 \\ 2 & -3 & -6 \\ \end{bmatrix}$ \]

   2. det(B)=1

      cof(B11)=${(-1)}^(i+j)$det(B11)=$(-1)^{(1+1)}((0*1)-(1*1))=-1$

      cof(B12)=${(-1)}^(i+j)$det(B12)=$(-1)^{(1+2)}((2*0)-(2*1))=2$

      cof(B13)=${(-1)}^(i+j)$det(B13)=$(-1)^{(1+3)}((2*1)-(2*1))=0$

      cof(B21)=${(-1)}^(i+j)$det(B21)=$(-1)^{(2+1)}((-1*0)-(1*-3))=-3$

      cof(B22)=${(-1)}^(i+j)$det(B22)=$(-1)^{(2+2)}((-3*0)-(2*-3))=6$

      cof(B23)=${(-1)}^(i+j)$det(B23)=$(-1)^{(2+3)}((-3*1)-(2*-1))=1$

      cof(B31)=${(-1)}^(i+j)$det(B31)=$(-1)^{(3+1)}((-1*1)-(1*-3))=2$

      cof(B32)=${(-1)}^(i+j)$det(B32)=$(-1)^{(3+2)}((-3*1)-(2*-3))=-3$

      cof(B33)=${(-1)}^(i+j)$det(B33)=$(-1)^{(3+3)}((-3*1)-(2*-1))=-1$

      \[ inv(B)=\frac{1}{det(A)}\mbox{adj}(B)=\frac{1}{\det(B)}, \quad M(B)'=1 \begin{bmatrix} -1 & 2 & 0 \\ -3 & 6 & 1 \\ 2 & -3 & -1 \\ \end{bmatrix}' =1\begin{bmatrix} -1 & -3 & 2 \\ 2 & 6 & -3 \\ 0 & 1 & -1 \\ \end{bmatrix} =\begin{bmatrix} -1 & -3 & 2 \\ 2 & 6 & -3 \\ 0 & 1 & -1 \\ \end{bmatrix} \]

Section 7: Elimination: A = LU

1. 1. \[ \mbox{(a)} \quad \begin{bmatrix} 1&0&0&0 \\ 3&1&0&0 \\ -1&0&1&0 \\ -3&4&-2&1 \end{bmatrix} \begin{bmatrix} 1&-2&-2&-3 \\ 0&-3&6&0 \\ 0&0&2&4 \\ 0&0&0&1 \end{bmatrix} , \quad \mbox{(b)} \quad \]

    

   Section 8: PLU Factorization

   1. 1. \[ \mbox{(a)} \quad \begin{bmatrix} 1&0&0&0 \\ 3/4&1&0&0 \\ 1/2&-2/7&1&0 \\ 1/4&-3/7&1/3&1 \end{bmatrix} \begin{bmatrix} 8&7&9&5 \\ 0&7/4&9/4&17/4 \\ 0&0&-6/7&-2/7 \\ 0&0&0&2/3 \end{bmatrix} , \quad \begin{bmatrix} 0&0&1&0 \\ 0&0&0&1 \\ 0&1&0&0 \\ 1&0&0&0 \end{bmatrix} , \] \[ \mbox{(b)} \quad \begin{bmatrix} 0&1&0&0 \\ 1&0&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} ,

       

      Section 9: Reflection

       

      Section 10: Givens Rotation

       

      Section 11: Special Matrices