Solutions to Problem Set 3

Math 311-01/02

1-29-2018

[Matrices]

1. Compute $\displaystyle 6 \cdot
   \left[\matrix{-3 & 6 \cr -1 & -5 \cr}\right] + \left[\matrix{0 & 1
   \cr -4 & 4 \cr}\right]$ .

$$6 \cdot \left[\matrix{-3 & 6 \cr -1 & -5 \cr}\right] + \left[\matrix{0 & 1 \cr -4 & 4 \cr}\right] = \left[\matrix{-18 & 37 \cr -10 & -26 \cr}\right].\quad\halmos$$


2. Compute $\displaystyle -4 \cdot
   \left[\matrix{-4 & -2 & -1 \cr 4 & -3 & 1 \cr}\right] +
   \left[\matrix{4 & 0 & 0 \cr -2 & 4 & 2 \cr}\right]$ .

$$-4 \cdot \left[\matrix{ -4 & -2 & -1 \cr 4 & -3 & 1 \cr}\right] + \left[\matrix{ 4 & 0 & 0 \cr -2 & 4 & 2 \cr}\right] = \left[\matrix{ 20 & 8 & 4 \cr -18 & 16 & -2 \cr}\right].\quad\halmos$$


3. Compute $\displaystyle
   \left[\matrix{0 & 1 \cr -3 & -6 \cr}\right] \left[\matrix{2 & 4 \cr
   -6 & -1 \cr}\right]$ .

$$\left[\matrix{0 & 1 \cr -3 & -6 \cr}\right] \left[\matrix{2 & 4 \cr -6 & -1 \cr}\right] = \left[\matrix{-6 & -1 \cr 30 & -6 \cr}\right].\quad\halmos$$


4. Compute $\displaystyle
   \left[\matrix{0 & -5 & 4 \cr}\right] \left[\matrix{-6 \cr -4 \cr 2
   \cr}\right]$ .

$$\left[\matrix{0 & -5 & 4 \cr}\right] \left[\matrix{-6 \cr -4 \cr 2 \cr}\right] = \left[\matrix{28 \cr}\right].\quad\halmos$$


5. Compute $\displaystyle
   \left[\matrix{5 \cr 3 \cr}\right] \left[\matrix{-1 & -2 \cr}\right]$ .

$$\left[\matrix{5 \cr 3 \cr}\right] \left[\matrix{-1 & -2 \cr}\right] = \left[\matrix{-5 & -10 \cr -3 & -6 \cr}\right].\quad\halmos$$


6. Compute $\displaystyle
   \left[\matrix{2 & -1 \cr -5 & -3 \cr}\right] \left[\matrix{-5 & -1
   \cr 4 & 4 \cr}\right] + \left[\matrix{0 & -2 \cr 2 & 1 \cr}\right]$ .

$$\left[\matrix{2 & -1 \cr -5 & -3 \cr}\right] \left[\matrix{-5 & -1 \cr 4 & 4 \cr}\right] + \left[\matrix{0 & -2 \cr 2 & 1 \cr}\right] = \left[\matrix{-14 & -6 \cr 13 & -7 \cr}\right] + \left[\matrix{0 & -2 \cr 2 & 1 \cr}\right] = \left[\matrix{-14 & -8 \cr 15 & -6 \cr}\right].\quad\halmos$$


7. Compute $\displaystyle
   \left[\matrix{3 & -8 \cr 4 & 0 \cr 1 & 11 \cr}\right] \left[\matrix{x
   \cr y \cr}\right]$ .

$$\left[\matrix{3 & -8 \cr 4 & 0 \cr 1 & 11 \cr}\right] \left[\matrix{x \cr y \cr}\right] = \left[\matrix{3 x - 8 y \cr 4 x \cr x + 11 y \cr}\right].\quad\halmos$$


[Determinants]

8. Compute the determinant: $\displaystyle \left|\matrix{3 & -5 \cr 1 & 9 \cr}\right|$ .

$$\left|\matrix{3 & -5 \cr 1 & 9 \cr}\right| = (3)(9) - (-5)(1) = 32.\quad\halmos$$


9. Compute the determinant: $\displaystyle \left|\matrix{-3 & 6 \cr 2 & 5 \cr}\right|$ .

$$\left|\matrix{-3 & 6 \cr 2 & 5 \cr}\right| = (-3)(5) - (6)(2) = -27.\quad\halmos$$


10. Compute the following determinant:

$$\left|\matrix{ 3 & 2 & 0 \cr 1 & 1 & -1 \cr 2 & 1 & 2 \cr}\right|.$$

I'll expand by cofactors using row 1:

$$\left|\matrix{ 3 & 2 & 0 \cr 1 & 1 & -1 \cr 2 & 1 & 2 \cr}\right| = 3 \cdot \left|\matrix{1 & -1 \cr 1 & 2 \cr}\right| - 2 \cdot \left|\matrix{1 & -1 \cr 2 & 2 \cr}\right| + 0 \cdot \left|\matrix{1 & 1 \cr 2 & 1 \cr}\right| = (3)(3) - (2)(4) + (0)(-1) = 1.\quad\halmos$$


11. Compute the following determinant:

$$\left|\matrix{ 4 & 1 & -5 \cr 1 & 0 & 3 \cr 1 & 2 & 1 \cr}\right|.$$

I'll expand by cofactors using column 2:

$$\left|\matrix{ 4 & 1 & -5 \cr 1 & 0 & 3 \cr 1 & 2 & 1 \cr}\right| = -1 \cdot \left|\matrix{1 & 3 \cr 1 & 1 \cr}\right| + 0 \cdot \left|\matrix{4 & -5 \cr 1 & 1 \cr}\right| - 2 \cdot \left|\matrix{4 & -5 \cr 1 & 3 \cr}\right| = -(1)(-2) + (0)(9) - (2)(17) = -32.\quad\halmos$$


12. Compute the following determinant:

$$\left|\matrix{ 1 & 1 & 1 \cr 3 & 1 & -1 \cr 2 & -1 & 2 \cr}\right|.$$

I'll expand by cofactors using row 1:

$$\left|\matrix{ 1 & 1 & 1 \cr 3 & 1 & -1 \cr 2 & -1 & 2 \cr}\right| = 1 \cdot \left|\matrix{1 & -1 \cr -1 & 2 \cr}\right| - 1 \cdot \left|\matrix{3 & -1 \cr 2 & 2 \cr}\right| + 1 \cdot \left|\matrix{3 & 1 \cr 2 & -1 \cr}\right| = (1)(1) - (1)(8) + (1)(-5) = -12.\quad\halmos$$


13. Suppose that

$$\det \left[\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr}\right] = 12.$$

Find:

(a) $\displaystyle \det
   \left[\matrix{5 a & 5 b & 5 c \cr d & e & f \cr 2 g & 2 h & 2 i
   \cr}\right]$ .

(b) $\displaystyle \det
   \left[\matrix{a & b & c \cr g & h & i \cr 3 d & 3 e & 3 f
   \cr}\right]$ .

(c) $\displaystyle \det
   \left[\matrix{a & b & c \cr a + 2 d & b + 2 e & c + 2 f \cr g & h & i
   \cr}\right]$ .

(a)

$$\det \left[\matrix{ 5 a & 5 b & 5 c \cr d & e & f \cr 2 g & 2 h & 2 i \cr}\right] = 5 \cdot \det \left[\matrix{ a & b & c \cr d & e & f \cr 2 g & 2 h & 2 i \cr}\right] = 10 \cdot \det \left[\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr}\right] = 10 \cdot 12 = 120.\quad\halmos$$

(b)

$$\det \left[\matrix{ a & b & c \cr g & h & i \cr 3 d & 3 e & 3 f \cr}\right] = 3 \cdot \det \left[\matrix{ a & b & c \cr g & h & i \cr d & e & f \cr}\right] = -3 \cdot \det \left[\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr}\right] = (-3) \cdot 12 = -36.\quad\halmos$$

(c)

$$\det \left[\matrix{ a & b & c \cr a + 2 d & b + 2 e & c + 2 f \cr g & h & i \cr}\right] = \det \left[\matrix{ a & b & c \cr a & b & c \cr g & h & i \cr}\right] + \det \left[\matrix{ a & b & c \cr 2 d & 2 e & 2 f \cr g & h & i \cr}\right] =$$

$$0 + 2 \cdot \det \left[\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr}\right] = 2 \cdot 12 = 24.\quad\halmos$$


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