Deck 12: Matrices

Question 1

Consider $A=\left(\begin{array}{cccc}3 & -1 & 1 & 0 \\ 2 & 0 & 3 & -1 \\ 1 & 0 & -2 & 1\end{array}\right)$ .
(a) Is $A$ a rectangular array?
(b) Is $A$ a triangular array?

Answer 1

(a) yes
(b) no

Question 2

Consider $A=\left(\begin{array}{cccc}3 & -1 & 1 & 0 \\ 2 & 0 & 3 & -1 \\ 1 & 0 & -2 & 1\end{array}\right)$ .
(a) Is $A$ a column vector?
(b) Is $A$ a row vector?
(c) Is $A$ a table?
(d) Is $A$ a list?
(e) Is $A$ a scalar?

Answer 2

(a) no
(b) no
(c) yes
(d) no
(e) no

Question 3

Consider $A=\left(\begin{array}{cccc}3 & -1 & 1 & 0 \\ 2 & 0 & 3 & -1 \\ 1 & 0 & -2 & 1\end{array}\right)$ .
(a) Is $A$ a null matrix?
(b) Is $A$ a matrix?

Answer 3

(a) no
(b) yes

Question 4

What are the elements $a_{23}$ and $a_{34}$ for $A=\left(\begin{array}{cccc}3 & -1 & 1 & 0 \\ 2 & 0 & 3 & -1 \\ 1 & 0 & -2 & 1\end{array}\right)$ ?

Question 5

What are the dimensions of $A=\left(\begin{array}{cccc}3 & -1 & 1 & 0 \\ 2 & 0 & 3 & -1 \\ 1 & 0 & -2 & 1\end{array}\right)$ ?

Question 6

Write the transpose of $A=\left(\begin{array}{cccc}3 & -1 & 1 & 0 \\ 2 & 0 & 3 & -1 \\ 1 & 0 & -2 & 1\end{array}\right)$ .

Question 7

Consider $B=\left(\begin{array}{lll}3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4\end{array}\right)$ .
(a) Is $B$ a square matrix?
(b) Is $B$ a diagonal matrix?
(c) Is $B$ a scalar matrix?

Question 8

Consider $B=\left(\begin{array}{lll}3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4\end{array}\right)$ .
(a) Is $B$ a column vector?
(b) Is $B$ a row vector?
(c) Is $B$ a table?
(d) Is $B$ a list?
(e) Is $B$ a scalar?

Question 9

What are elements $b_{22}$ and $b_{13}$ for $B=\left(\begin{array}{lll}3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4\end{array}\right)$ ?

Question 10

Write the transpose of $B=\left(\begin{array}{lll}3 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 4\end{array}\right)$ .

Question 11

What are the dimensions of $C=\left(\begin{array}{lll}3 & 2 & 4 \\ 8 & 1 & 5 \\ 2 & 2 & 4 \\ 9 & 6 & 7\end{array}\right)$ ?

Question 12

Write the transpose of $C=\left(\begin{array}{lll}3 & 2 & 4 \\ 8 & 1 & 5 \\ 2 & 2 & 4 \\ 9 & 6 & 7\end{array}\right)$ .

Question 13

What are the elements $c_{32}$ and $c_{23}$ for $C=\left(\begin{array}{lll}3 & 2 & 4 \\ 8 & 1 & 5 \\ 2 & 2 & 4 \\ 9 & 6 & 7\end{array}\right)$ ?

Question 14

Write the transpose of $C=\left(\begin{array}{lll}6 & 0 & 4 \\ 0 & 1 & 1 \\ 4 & 1 & 4\end{array}\right)$ .

Question 15

Under what conditions will these matrices be equal? $\left(\begin{array}{ccc}-2 & a & 2 \\ 3 & b & -1\end{array}\right)$ and $\left(\begin{array}{ccc}-2 & -1 & c \\ 3 & 5 & d\end{array}\right)$

Question 16

Under what conditions will these matrices be equal? $\left(\begin{array}{cc}a & -1 \\ 2 & b\end{array}\right)$ and $\left(\begin{array}{cc}-3 & c \\ d & 4\end{array}\right)$

Question 17

Calculate: $\left(\begin{array}{cccc}-2 & 1 & -3 & 4 \\ 1 & 0 & 2 & -3\end{array}\right)-\left(\begin{array}{cccc}3 & 4 & -1 & 2 \\ 5 & -1 & 3 & -2\end{array}\right)$

Question 18

Calculate: $\left(\begin{array}{llll}2 & 4 & 7 & 3\end{array}\right)+\left(\begin{array}{llll}11 & 9 & -3 & 5\end{array}\right)$

Question 19

Calculate: $\left(\begin{array}{ccc}0 & -1 & 3 \\ 1 & 2 & 4 \\ -1 & 0 & -2\end{array}\right)-\left(\begin{array}{ccc}2 & 0 & 3 \\ -1 & 0 & 3 \\ -2 & -1 & 1\end{array}\right)$

Question 20

Calculate: $\left(\begin{array}{ccc}5 & 3 & -4 \\ 7 & 2 & 6 \\ 5 & 6 & 9\end{array}\right)-\left(\begin{array}{ccc}2 & 8 & 3 \\ 5 & 6 & -2 \\ 1 & 7 & 4\end{array}\right)+\left(\begin{array}{ccc}-1 & -6 & 5 \\ 3 & 8 & 2 \\ 7 & -2 & 6\end{array}\right)$

Question 21

Calculate: $\left(\begin{array}{c}4 \\ -1 \\ 2\end{array}\right)+\left(\begin{array}{l}3 \\ 6 \\ 2\end{array}\right)$

Question 22

Multiply: $7\left(\begin{array}{lll}6 & 3 & 1 \\ 9 & 2 & 5\end{array}\right)$

Question 23

Multiply: $-4\left(\begin{array}{cccc}3 & -2 & 6 & 5 \\ -3 & 7 & 2 & 9\end{array}\right)$

Question 24

Calculate: $3\left(\begin{array}{cc}1 & 2 \\ -4 & 2\end{array}\right)+2\left(\begin{array}{cc}1 & 0 \\ 0 & 3\end{array}\right)-\left(\begin{array}{cc}4 & 2 \\ 4 & 1\end{array}\right)$

Question 25

Remove the greatest factor possible from this matrix: $\left(\begin{array}{cc}12 & -4 \\ 8 & 24\end{array}\right)$

Question 26

Remove the greatest factor possible from this vector: $\left(\begin{array}{c}-6 \\ 9 \\ 51\end{array}\right)$

Question 27

Multiply: $\left(\begin{array}{ll}7 & 3\end{array}\right)\left(\begin{array}{c}-2 \\ 5\end{array}\right)$

Question 28

Multiply: $\left(\begin{array}{cc}2 & 1 \\ 7 & -2\end{array}\right)\left(\begin{array}{c}3 \\ -2\end{array}\right)$

Question 29

Multiply: $\left(\begin{array}{ccc}2 & 9 & 2 \\ 0 & 3 & 6 \\ 4 & 1 & -1\end{array}\right)\left(\begin{array}{c}3 \\ 6 \\ -7\end{array}\right)$

Question 30

Multiply: $\left(\begin{array}{lll}4 & -5 & 3\end{array}\right)\left(\begin{array}{c}3 \\ 6 \\ -7\end{array}\right)$

Question 31

Multiply: $\left(\begin{array}{ll}3 & 2\end{array}\right)\left(\begin{array}{cccc}2 & 4 & 3 & -7 \\ 1 & 2 & 1 & 9\end{array}\right)$

Question 32

Multiply: $\left(\begin{array}{lll}4 & 8 & 3\end{array}\right)\left(\begin{array}{cccc}2 & 8 & -3 & 2 \\ 1 & 4 & 9 & -4 \\ 6 & 7 & 5 & -7\end{array}\right)$

Question 33

Multiply: $\left(\begin{array}{cc}1 & -2 \\ 0 & 4\end{array}\right)\left(\begin{array}{ccc}7 & 2 & 5 \\ 1 & -2 & 3\end{array}\right)$

Question 34

Multiply: $\left(\begin{array}{cc}7 & 1 \\ -4 & 5 \\ 6 & 0\end{array}\right)\left(\begin{array}{cc}-6 & -4 \\ 2 & 7\end{array}\right)$

Question 35

Multiply: $\left(\begin{array}{ccc}1 & 2 & -1 \\ 3 & 4 & 0\end{array}\right)\left(\begin{array}{cc}-1 & -2 \\ 3 & 0 \\ -4 & -1\end{array}\right)$

Question 36

Multiply: $\left(\begin{array}{lll}2 & 8 & 7 \\ 6 & 2 & 1\end{array}\right)\left(\begin{array}{cc}-1 & 3 \\ 5 & 4 \\ 9 & 0\end{array}\right)$

Question 37

Multiply: $\left(\begin{array}{ccc}3 & -4 & 7 \\ 2 & 5 & 8 \\ 1 & -6 & 9\end{array}\right)\left(\begin{array}{c}-3 \\ 2 \\ -1\end{array}\right)$

Question 38

Multiply: $\left(\begin{array}{llll}5 & 6 & 7 & 8 \\ 1 & 2 & 4 & 5 \\ 4 & 5 & 6 & 7\end{array}\right)\left(\begin{array}{cccc}-8 & -2 & -1 & 2 \\ 7 & 3 & 3 & -4 \\ -6 & -4 & -5 & 6 \\ 5 & 5 & 7 & -8\end{array}\right)$

Question 39

Multiply: $\left(\begin{array}{ccc}5 & -3 & 0 \\ 2 & 1 & 7 \\ 0 & 5 & 4\end{array}\right)\left(\begin{array}{cccc}6 & 0 & 4 & -5 \\ 1 & 3 & 2 & -1 \\ -4 & -2 & 1 & 9\end{array}\right)$

Question 40

Calculate $A B$ and $B A$ for $A=\left(\begin{array}{cc}4 & -3 \\ 8 & 1\end{array}\right)$ and $B=\left(\begin{array}{cc}5 & 0 \\ 6 & -2\end{array}\right)$ .

Question 41

Calculate $A B$ and $B A$ for $A=\left(\begin{array}{cc}-a & 2 \\ 3 & 3 a\end{array}\right)$ and $B=\left(\begin{array}{cc}2 & -a \\ 3 a & -1\end{array}\right)$ .

Question 42

Calculate $A B$ and $B A$ for $A=\left(\begin{array}{lll}1 & 0 & 3 \\ 4 & 2 & 0 \\ 0 & 0 & 2\end{array}\right)$ and $B=\left(\begin{array}{ccc}2 & 3 & 0 \\ 1 & 1 & -1 \\ 6 & 0 & 8\end{array}\right)$ .

Question 43

If $A=\left(\begin{array}{ccc}3 & 4 & -1 \\ -5 & 1 & 7 \\ 2 & 3 & 1\end{array}\right)$ and $B=\left(\begin{array}{ccc}-2 & 3 & 9 \\ 0 & -4 & -2 \\ 5 & 6 & 3\end{array}\right)$ , calculate $2 A-B$ .

Question 44

If $A=\left(\begin{array}{ccc}3 & 4 & -1 \\ -5 & 1 & 7 \\ 2 & 3 & 1\end{array}\right)$ and $B=\left(\begin{array}{ccc}-2 & 3 & 9 \\ 0 & -4 & -2 \\ 5 & 6 & 3\end{array}\right)$ , calculate $3 A+2 B$ .

Question 45

Three types of seating were available for a concert at an arena, with tickets priced at $\$ 45.99, \$ 52.45$ , and $\$ 57.00$ . One day the arena sold 75,48 , and 66 of these tickets, respectively. Use vector multiplication to determine the total amount paid for the tickets that day.

Question 46

Multiply to show that $\left(\begin{array}{cc}4 & -2 \\ -1 & 3\end{array}\right)$ and $\left(\begin{array}{ll}0.3 & 0.2 \\ 0.1 & 0.4\end{array}\right)$ are inverses.

Question 47

Multiply to show that $\left(\begin{array}{ccc}2 & 1 & 1 \\ -1 & -1 & 1 \\ 0 & 1 & 2\end{array}\right)$ and $\left(\begin{array}{ccc}0.6 & 0.2 & -0.4 \\ -0.4 & -0.8 & 0.6 \\ 0.2 & 0.4 & 0.2\end{array}\right)$ are inverses.

Question 48

Calculate the inverse: $\left(\begin{array}{cc}-1 & -2 \\ 1 & -1\end{array}\right)$

Question 49

Calculate the inverse: $\left(\begin{array}{cc}10 & -10 \\ 15 & -5\end{array}\right)$

Question 50

Calculate the inverse: $\left(\begin{array}{ll}2.0 & 2.0 \\ 4.0 & 5.0\end{array}\right)$

Question 51

Calculate the inverse: $\left(\begin{array}{ll}1 & -4 \\ 1 & -5\end{array}\right)$

Question 52

Calculate the inverse of $A=\left(\begin{array}{cc}8 & 3 \\ -2 & 6\end{array}\right)$ . Work to three decimal places.

Question 53

Calculate the inverse of $A=\left(\begin{array}{cc}5 & -7 \\ 1 & 4\end{array}\right)$ . Work to three decimal places.

Question 54

Calculate the inverse of $A=\left(\begin{array}{ll}2 & 5 \\ 6 & 3\end{array}\right)$ . Work to three decimal places.

Question 55

Calculate the inverse: $\left(\begin{array}{ccc}-1 & 1 & 5 \\ 2 & 1 & 0 \\ 4 & 1 & -2\end{array}\right)$ . Work to two decimal places.

Question 56

Calculate the inverse: $\left(\begin{array}{lll}7.0 & 4.0 & 1.0 \\ 5.0 & 3.0 & 3.0 \\ 8.0 & 4.0 & 2.0\end{array}\right)$

Question 57

Calculate the inverse of $A=\left(\begin{array}{lll}1 & 4 & 2 \\ 6 & 0 & 3 \\ 8 & 1 & 5\end{array}\right)$ . Work to three decimal places.

Question 58

Calculate the inverse of $A=\left(\begin{array}{ccc}6 & 2 & -7 \\ -4 & 9 & 1 \\ 3 & 5 & -3\end{array}\right)$ . Work to three decimal places.

Question 59

Calculate the inverse: $\left(\begin{array}{ccc}2 & -1 & -2 \\ 4 & 2 & 1 \\ 2 & 1 & -1\end{array}\right)$

Question 60

Calculate the inverse: $\left(\begin{array}{ccc}3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2\end{array}\right)$

Question 61

Calculate the inverse: $\left(\begin{array}{ccc}-15 & 4 & -5 \\ 12 & -3 & 4 \\ 4 & -1 & 1\end{array}\right)$

Question 62

Calculate the inverse: $\left(\begin{array}{ccc}1 & 1 & 1 \\ 1 & 5 & 0 \\ 0 & 1 & -3\end{array}\right)$

Question 63

Calculate the inverse: $\left(\begin{array}{ccc}-5 & 3 & 0 \\ -2 & 1 & 0 \\ 0 & 1 & 1\end{array}\right)$

Question 64

Calculate the inverse: $\left(\begin{array}{ccc}-3 & -3 & 1 \\ 2 & 2 & -1 \\ 4 & 5 & -2\end{array}\right)$

Question 65

Calculate the inverse of $A=\left(\begin{array}{cccc}-3 & 0 & 6 & -7 \\ 6 & 1 & 3 & -6 \\ -5 & 2 & 5 & 1 \\ 4 & 4 & 8 & 3\end{array}\right)$ . Work to three decimal places.

Question 66

Calculate the inverse: $\left(\begin{array}{cccc}2 & -1 & 3 & 0 \\ -1 & 2 & -2 & 4 \\ 1 & 0 & 1 & 2 \\ -2 & 1 & -3 & 1\end{array}\right)$

Question 67

Calculate the inverse of $A=\left(\begin{array}{cccc}1 & 1 & 2 & 3 \\ 0 & 1 & -2 & 5 \\ 5 & 4 & 0 & 2 \\ 2 & 1 & 2 & 2\end{array}\right)$ . Work to three decimal places.

Question 68

Solve the system of equations by matrix inversion: $x-2 y=1$
$2 x+5 y=20$

Question 69

Solve the system of equations by matrix inversion: $4 x+3 y=-9$
$3 x-5 y=-43$

Question 70

Solve the system of equations by matrix inversion: $5 x-4 y=8$
$x-y=-1$

Question 71

Solve the system of equations by matrix inversion: $x-4 y=4$
$x-5 y=3$

Question 72

Solve the system of equations by matrix inversion: $3 x-4 y=26$
$-x+2 y=-12$

Question 73

Solve the system of equations by matrix inversion: $2 x+11 y=-8$
$3 x+4 y=13$

Question 74

Solve the system of equations by matrix inversion: $3 x+11 y=50$
$3 x+5 y=26$

Question 75

Solve the system of equations by matrix inversion: $-x+4 y=5$
$3 x+5 y=2$

Question 76

Solve the system of equations by matrix inversion: $x+15 y=85$
$x+4 y=19$

Question 77

Solve the system of equations by matrix inversion: $6 x+12 y=48$
$5 x+20 y=60$

Question 78

Solve the system of equations by matrix inversion: $2 x-10 y=5$
$8 x+2 y=6$

Question 79

Solve the system of equations by matrix inversion: $x+y+z=3$
$\begin{aligned}& x-y-3 z=1 \\& x+y+2 z=2\end{aligned}$

Question 80

Solve the system of equations by matrix inversion: $4 x+9 y+2 z=33$
$\begin{aligned}3 x-5 y-3 z & =-6 \\x+y+2 z & =3\end{aligned}$