Mathematical Immaturity

3.11 $\quad$ Exercises

• For each of the following statements about square matrices, give a proof or exhibit a counter example.
  (a) $\quad$ $\det (A + B) = \det A + \det B.$
  (b) $\quad$ $\det \{(A + B)^2\} = \{\det (A + B)\}^2$
  (c) $\quad$ $\det \{(A + B)^2\} = \det (A^2 + 2AB + B^2)$
  (d) $\quad$ $\det \{(A + B)^3\} = \det (A^3 + B^3).$
• (a) $\quad$ Extend Theorem 3.7 to block-diagonal matrices with three diagonal blocks: $$\det \begin{bmatrix} A & O & O \\ O & B & O \\ O & O & C \end{bmatrix} = (\det A)(\det B)(\det C).$$
  (b) $\quad$ State and prove a generalization for block-diagonal matrices with any number of diagonal blocks.
• Let $A = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ a & b & c & d \\ e & f & g & h \end{bmatrix},$ $B = \begin{bmatrix} a & b & c & d \\ e & f & g & h \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}.$ Prove that $\det A = \det \begin{bmatrix} c & d \\ g & h \end{bmatrix}$ and that $\det B = \det \begin{bmatrix} a & b \\ e & f \end{bmatrix}.$
• State and prove a generalization of Exercise 3 for $n \times n$ matrices.
• Let $A = \begin{bmatrix} a & b & 0 & 0 \\ c & d & 0 & 0 \\ e & f & g & h \\ x & y & z & w \end{bmatrix}.$ Prove that $\det A = \det \begin{bmatrix} a & b \\ c & d \end{bmatrix} \det \begin{bmatrix} g & h \\ z & w \end{bmatrix}.$
• State and prove a generalization of Exercise 5 for $n \times n$ matrices of the form $$A = \begin{bmatrix} B & O \\ C & D \end{bmatrix}$$ where $B,$ $C,$ $D$ denote square matrices and $O$ denotes a matrix of zeros.
• Use Theorem 3.6 to determine whether the following sets of vectors are linearly dependent or independent.
  (a) $\quad$ $A_1 = (1, -1, 0),$ $A_2 = (0, 1, -1),$ $A_3 = (2, 3, -1).$
  (b) $\quad$ $A_1 = (1, -1, 2, 1),$ $A_2 = (-1, 2, -1, 0),$ $A_3 = (3, -1, 1, 0),$ $A_4 = (1, 0, 0, 1).$
  (c) $\quad$ $A_1 = (1, 0, 0, 0, 1),$ $A_2 = (1, 1, 0, 0, 0),$ $A_3 = (1, 0, 1, 0, 1),$ $A_4 = (1, 1, 0, 1, 1),$ $A_5 = (0, 1, 0, 1, 0).$