Math210E Linear Systems and Matrices 1

Which are the following matrices in row echelon form? If not, explain why.
       
1 5 12 1 3 2 7 1 3 −5 17 1 0 1
a)  0 1 5  b)  0 0 1 7  c)  0 1 1 7  d)  0 0 0 
       

0 0 1 0 1 9 −1 0 0 0 1 0 0 1

   
1 2 3  1 0 5 
 1 4 1 .
Find the reduced echelon form of the matrix   
 0 1 −1 

2 1 9 0 0 0

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Math210E Linear Systems and Matrices 2

Find the reduced echelon form of each of the matrices.

   
1 −4 −2  1 −4 0 
 3 −12 1   0 0 1 
   

2 −8 5 0 0 0

   
1 3 2 5  1 0 0 4 
 2 5 2 3   0 1 0 −3 
   

2 7 7 22 0 0 1 5

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Math210E Linear Systems and Matrices 3

Use elementary row operations to transform each augmented coefficient matrix to echelon form.
Then solve the system by back substitution.
2x − 4y + z = 3
x − 3y + z = 5 (inconsistent , no solution)
3x − 7y + 2z = 12

−3x − 2y + 4z = 9
3y − 2z = 5 (x = 3 , y = 7 , z = 8)
4x − 3y + 2z = 7

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Math210E Linear Systems and Matrices 4

Use elementary row operations to transform each augmented coefficient matrix to echelon form.
Then solve the system by back substitution.
2x1 + 8x2 + 3x3 = 2
x1 + 3x2 + 2x3 = 5 (x1 = 3 , x2 = −2 , x3 = 4)
2x1 + 7x2 + 4x3 = 8

2x1 + 5x2 + 12x3 = 6

3x1 + x2 + 5x3 = 12 (inconsistent , no solution)
5x1 + 8x2 + 21x3 = 17

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Math210E Linear Systems and Matrices 5

Use Gaussian elimination to solve the system.

x1 + 3x2 + 3x3 = 13
2x1 + 5x2 + 4x3 = 23 (x1 = 4+3t , x2 = 3−2t , x3 = t)
2x1 + 7x2 + 8x3 = 29

3x1 − 6x2 + x3 + 13x4 = 15

3x1 − 6x2 + 3x3 + 21x4 = 21 (x1 = 4 + 2s − 3t , x2 = s , x3 = 3 − 4t , x4 = t)
2x1 − 4x2 + 5x3 + 26x4 = 23

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Math210E Linear Systems and Matrices 6

Use Gaussian elimination to solve the system.

x1 + x2 + x3 = 6
2x1 − 2x2 − 5x3 = −13
(x1 = 2 , x2 = 1 , x3 = 3 , x4 = 4)
3x1 + x3 + x4 = 13
4x1 − 2x2 − 3x3 + x4 = 11

x1 − 4x2 − 3x3 − 3x4 = 4

2x1 − 6x2 − 5x3 − 5x4 = 5 (x1 = 3 − 2t , x2 = −4 + t , x3 = 5 − 3t , x4 = t)
3x1 − x2 − 4x3 − 5x4 = −7

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Math210E Linear Systems and Matrices 7

Use Gaussian elimination to solve the system.

2x1 + 4x2 − x3 − 2x4 + 2x5 = 6
x1 + 3x2 + 2x3 − 7x4 + 3x5 = 9
5x1 + 8x2 − 7x3 + 6x4 + x5 = 4
(x1 = 2 + 3t , x2 = 1 + s − 2t , x3 = 2 + 2s , x4 = s , x5 = t)

3x1 + x2 − 3x3 = −4
x1 + x2 + x 3 = 1 (inconsistent , no solution)
5x1 + 6x2 + 8x3 = 8

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Math210E Linear Systems and Matrices 8

Determine for what values of k each system has a unique solution, no solution or infinitely many
solutions.
3x + 2y = 11
(k 6= 4 , k = 4 , none)
6x + ky = 21

3x + 2y = 1
(all k , none , none)
7x + 5y = k

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Math210E Linear Systems and Matrices 9

Determine for what values of k each system has a unique solution, no solution or infinitely many
solutions.
x + 2y + z = 3
2x − y − 3z = 5 (none , k 6= 11 , k = 11)
4x + 3y − z = k

x + 2y + 2z = 4
2x 4 ky + 4z = 8 (k 6= 0 or 4 , k = 0 , k = 4)
kz = 4

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Math210E Linear Systems and Matrices 10

Use elementary row operations to transform each augmented coefficient matrix to reduced echelon
form. Then solve the system by back substitution.
x + 2y − z = 1
2x + y + 4z = 2 (x = 7 , y = −4 , z = −2)
3x + 3y + 4z = 1

x1 − x2 + x3 − x4 = 2
x1 − x2 + x3 + x4 = 0
(x1 = 1 + t − s , x2 = t , x3 = s , x4 = −1)
4x1 − 4x2 + 4x3 = 4
−2x1 + 2x2 − 2x3 + x4 = −3

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Math210E Linear Systems and Matrices 11

Use elementary row operations to transform each augmented coefficient matrix to reduced echelon
form. Then solve the system by back substitution.
3x + 9y + z = 16
2x + 6y + 7z = 3 (x = 5−t , y = t , z = 1)
x + 3y − 6z = −1

x1 + 3x2 + 15x3 7x4 = 1

2x1 + 4x2 + 22x3 + 8x4 = 2 (inconsistent)
2x1 + 7x2 + 34x3 + 17x4 = 3

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Math210E Linear Systems and Matrices 12

Determine for what values of α each system has a unique solution, no solution or infinitely many
solutions. In the case of infinitely many solutions, find the solution set.
x + y − z = 1
2x + 3y + αz = 3 (α 6= 2 and α 6= −3 , α = −3 , α = 2 , (5t, 1 − 4t, t))
x + αy + 3z = 2

x + 4y − 7z = 1
−x − 3y + 5z = 3 (α 6= 3 and α 6= −3 , α = −3 , α = 3 , (−t, 2 + 2t, t))
2
2x + 5y + (α − 17)z = α + 7

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Math210E Linear Systems and Matrices 13

Determine for what values of λ and β each system has a unique solution, no solution or infinitely
many solutions. In the case of unique and infinitely many solutions, find the solution sets.
x − βz = 1
−2x + λy − βz = −1
−x + λy βz = 0
(λ 6= 0 and β 6= 0 , λ = 0 , λ 6= 0 and β = 0 , (1, 1/a, 0) , (1, 1/(2a − 1), t))

x + 2y + 5z = 6
y + 2z = 2
−2x + 2y + (β − 1)z = 0
2x + 2y + 6z = λ+2
(λ = 6 and β 6= 3 , λ 6= 6 , λ = 6 and β = 3 , (2, 2, 0) , (2−t, 2−2t, t))

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Math210E Linear Systems and Matrices 14

Determine for what values of λ and β each system has a unique solution, no solution or infinitely
many solutions. In the case of infinitely many solutions, find the solution set.
2 −1 2λ 1 x1 β
    
 2 0 2λ 1  x2   1 
=
    
2 −1 (2λ + 1) (λ + 1) x3 0
   
    
−2 1 (1 − 2λ) −2 x4 −β − 2
(λ 6= −1 , λ = −1 and β 6= 2 , λ = −1 and β = 2 , ((t − 3)/2, −1, t − 2, t))

1 0 1 −1 x1 β
    
 −2 −1 −4 2  x2   −2 
  
= 
−1 −2 −3 λ−1 x3 1
    
    
−1 1 2λ − 3 1 x4 β−2
(λ 6= 2 , λ = 2 and β 6= 5 , λ = 2 and β = 5 , (2 − r + s, 1 − 2r, r, s))

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Math210E Linear Systems and Matrices 15

!
a b
Show that the 2 × 2 matrix A = is row equivalent to the 2 × 2 identity matrix provided
c d
that ad − bc 6= 0.

!
a b
If A = , then show that A2 = (traceA)A − (ad − bc)I where I is the 2 × 2 identity
c d
matrix.

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Math210E Linear Systems and Matrices 16

Compute the matrix cA + dB.

! ! !
2 0 −3 −2 3 1 16 −9 −18

A= , B= , c = 5, d = −3
−1 5 6 7 1 5 −26 22 15

     
2 −1 0 6 −3 −4  44 −22 −20 
 4 0 −3 ,
A= B =  5 2 −1 , c = 7, d=5  53 10 −26 
    

5 −2 7 0 7 9 35 21 94

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Math210E Linear Systems and Matrices 17

Compute the matries AB and BA if defined.

! ! ! !
2 −1 −4 2 −9 −1 −2 8

A= , B= ,
3 2 1 3 −10 12 11 5

! ! !
2 1 −1 0 4 1 −2 −13
 
A= , B= , not defined
4 3 3 −2 5 5 −6 3

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Math210E Linear Systems and Matrices 18

Compute the matries AB and BA if defined.

   
  3   3 6 9 
A= 1 2 3 , B=
 4 

26 ,  4 8 12 
 

5 5 10 15

   
! 0 −2 4
3
 
A= , B= 3

1  not defined ,  7 
 
−2

−4 5 −22

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Math210E Linear Systems and Matrices 19

Verify both sides of the associative law.

 
! 2 0 !
3 12 15
  
A= , B= 1 −1 2 , C= 0 3 
 
2 8 10
1 4

   
2 0 ! ! −4 −4 −2 2 
1 −1 1 0 −1 2

A= 0 3 
, B= , C=  −9 −12 9 12 
 
3 −2 3 2 0 1

1 4 −14 −18 −13 17

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Math210E Linear Systems and Matrices 20

Write each given homogeneous system in the matrix form Ax = 0. Then find the solution in
vector form.
x1 − 5x3 + 4x4 = 0
(x = s(5, −2, 1, 0) + t(−4, 7, 0, 1))
x2 + 2x3 − 7x4 = 0

x1 + 3x4 − x5 = 0
x2 − 2x4 + 6x5 = 0 (x = s(−3, 2, −1, 1, 0)+t(1, −6, 8, 0, 1))
x3 + x4 − 8x5 = 0

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Math210E Linear Systems and Matrices 21

Write each given homogeneous system in the matrix form Ax = 0. Then find the solution in
vector form.
x1 − 3x2 + 6x4 = 0
(x = s(3, 1, 0, 0) + t(−6, 0, −9, 1))
x3 + 9x4 = 0

x1 − x2 + 7x4 + 3x5 = 0
(x = r(1, 1, 0, 0, 0)+s(−7, 0, 1, 1, 0)+t(−3, 0, 2, 0, 1))
x3 − x4 − 2x5 = 0

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Math210E Linear Systems and Matrices 22

Write all 2 × 2 elementary matrices.

   
1 2 −3 1 2 −3
Let B =  2 3 3 . Find elementary matrix E so that EB =  2 3 3 .
   

5 9 −6 3 5 0
 
 1 0 0 
 0 1 0 
 

−2 0 1

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Math210E Linear Systems and Matrices 23

       
2 4 −2 0 3 7 1 2 −1 2 3 −2
Let A =  4 5 1  , B1 =  4 5 1  , B2 =  4 5 1  , B3 =  0 −3 5 .
       

0 3 7 2 4 −2 0 3 7 0 3 7
Find elementary matrices E1 , E2 and E3 such that E1 A = B1 , E2 A = B2 and E3 A = B3 .
Determine the inverses of E1 , E2 and E3 .
           
 0 0 1 1/2 0 0 1 0 0 0 0 1 2 0 0 1 0 0 

 0 1 0 ,
 
0 1 0 ,
 
−2 1 0 ,
 
0 1 0 ,
 
0 1 0 , 2 1 0 
  

1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1

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Math210E Linear Systems and Matrices 24

Find a 2 × 2 matrix A with each main diagonal element zero such that A2 = I.

Find a 2 × 2 matrix A with each main diagonal element zero such that A2 = −I.

   
0 1 1 1 1 0
3 7
Show taht there is no 3 × 3 matrix A such that A =  0 0 1  and A =  1 0 0 .
   

0 0 0 0 0 0

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Math210E Linear Systems and Matrices 25

Use matrix multiplication to show that if x1 and x2 are two solutions of the homogeneous system
Ax = 0 and c1 and c2 are real numbers, then c1 x1 + c2 x2 is also a solution.

Use matrix multiplication to show that if x0 is a solution of the homogeneous system Ax = 0

and x1 is a solution of the nonhomogeneous system Ax = b, then x0 + x1 is also a solution of the
nonhomogeneous system.

Use matrix multiplication to show that if x1 and x2 are solutions of the nonhomogeneous system
Ax = b, then x1 − x2 is a solution of the homogeneous system Ax = 0.

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Math210E Linear Systems and Matrices 26

Find A−1 of each given matrix A.

! !
5 7 1 6 −7


4 6 2 −4 5

   
1 5 1  −5 −2 5 
 2 5 0 
 
 2

1 −2 

2 7 1 −4 −3 5

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Math210E Linear Systems and Matrices 27

Find A−1 of each given matrix A.

   
1 −3 0 −2 −6 −3 
1

 −1 2 −1   −2 −2 −1 
  
4
0 −2 2 −2 −2 1

   
2 0 −1 3 1 0 
1

 1 0 3   −2 −3 7 
  
7
1 1 1 −1 2 0

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Math210E Linear Systems and Matrices 28

Calulate A−1 and then find a matrix X such that Ax = B.

! ! ! !
4 3 1 3 −5 4 −3 7 18 −35

−1
A= ,B = A = ,X =
5 4 −1 −2 5 −5 4 −9 −23 −45

       
1 4 1 1 0 3  11 −9 4 7 −14 15 
A =  2 8 3 ,B =  0 2 2 
  
 A−1 =  −2 2 −1  , X =  −1 3 −2 
  

2 7 4 −1 1 0 −2 1 0 −2 2 −4

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Math210E Linear Systems and Matrices 29

Calulate A−1 and then find a matrix X such that Ax = B.

! ! ! !
7 6 2 0 4 7 −6 14 −30 46

−1
A= ,B = A = ,X =
8 7 0 5 −3 −8 7 −16 35 −53

       
1 5 1 2 0 1  −16 3 11 −21 9 6 
A =  2 1 −2  , B =  0 3 0 
  
 A−1 =  6 −1 −4  , X =  8 −3 −2 
   

1 7 2 1 0 2 −13 2 9 −17 6 5

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Math210E Linear Systems and Matrices 30

! ! ! !
1 2 1 3 1 1 −1 2

Find 2 × 2 matrix A such that A = .
0 1 0 1 1 1 1 −2

 
1 1 1
Let A−1 = 1 1 2 

. Use elementary row operations to find A. Use A to solve the linear
1 −1 1
     
4  3/2 −1 1/2 −2 
−1
system A x = b where b =  7 .  1/2 0 −1/2  ,  3 
     

−2 −1 1 0 3

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Math210E Linear Systems and Matrices 31

Suppose that A, B, and C are invertible matrices of the same size. Show that the product ABC
is invertible and that (ABC)−1 = C −1 B −1 A−1 .

Let A be n n × n matrix such that Ax = x for every n-vector x. Show that A = I.

Prove that if A is invertible and AB = AC, then B = C.

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Math210E Linear Systems and Matrices 32

 
3 4 1
Let A =  1 −2 0 . Find the determinant of A by cofactor expansion (−47)
 

5 3 6
along the second column.

along the second row.

along the first column.

along the third row.

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Math210E Linear Systems and Matrices 33

Use cofactor expansions to evaluate the following determinants. Expand along the row or column
that minimizes the amount of computation that is required.
 

 2 1 0 

 1 2 1  (4)
 
 0 1 2 

 

 1 0 0 0 

 2 0 5 0 
 
(−210)
3 6 9 8
 
 
 
 4 0 10 7 

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Math210E Linear Systems and Matrices 34

Evaluate the determinants after first simplifying the computation by adding an appropriate
multiple of some row or column to another.
 

 2 3 4 


 −2 −3 1 
 (25)
 
 3 2 7 

 

 3 −2 5 

 0 5 17  (30)
6 −4 12 
 


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Math210E Linear Systems and Matrices 35

Use the method of elimination to evaluate the following determinants.

 

 2 3 3 1 
 0 4 3 −3 
(8)

2 −1 −1 −3 
 


 0 −4 −3 2 

 

 1 4 4 1 

 0 1 −2 2 
 
(135)
3 3 1 4
 
 
 
 0 1 −3 −2 

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Math210E Linear Systems and Matrices 36

Evaluate the following determinants.

 

 0 0 a1 b1 
 0 0 a2 b2 
((a2 b1 − a1 b2 )(a4 b4 − a3 b4 ))

a3 b3 0 0 
 


 a4 b 4 0 0 

 

 a1 0 0 b1 
 0 a2 b2 0 
((a1 a4 − b1 b4 )(a2 a3 − b2 b3 ))

0 b3 a3 0 
 


 b 4 0 0 a4 

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Math210E Linear Systems and Matrices 37

2 5 3 4
 
 −1 −2 −2 −3 
Let A =  .
 
 2 6 4 4 
1 3 8 9
Evaluate |A|. (2)

Find |AT |, |A−1 |, |A4 | and |2A|. (2 , 1/2 , 16 , 32)

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Math210E Linear Systems and Matrices 38

Use properties of determinants to show the following. Do not use the cofactor expansion.
x y + z y2 + z2
 
 
 

 y x + z x2 + z 2 
 = (x + y + z)(x − y)(x − z)(z − y).
z x + y x2 + y 2
 
 

 

 a b b b 

 b a b b 
= (a − b)3 (a + 3b).
 
b b a b
 
 
 
 b b b a 

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Math210E Linear Systems and Matrices 39

Use Cramer’s rule to solve the following systems.

5x1 + 2x2 − 2x3 = 1
x1 + 5x2 − 3x3 = −2 (x1 = 1/3 , x2 = −2/3 , x3 = −1/3)
5x1 − 3x2 + 5x3 = 2

3x1 − x2 − 5x3 = 3
4x1 − 4x2 − 3x3 = −4 (x1 = 2 , x2 = 3 , x3 = 0)
x1 − 5x3 = 2

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Math210E Linear Systems and Matrices 40

Use Cramer’s rule to solve the following systems.

2x1 + x2 − 3x3 = 0
4x1 + 5x2 + x3 = 8 (x1 = 4 , x2 = −2 , x3 = 2)
−2x1 − x2 + 4x3 = 2

3x1 + 2x2 − x3 = 1
x1 − 5x2 + 5x3 = −2 (x1 = 12 , x2 = −21 , x3 = −7)
2x1 + x2 = 3

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Math210E Linear Systems and Matrices 41

Use Cramer’s rule to solve the following systems.

2x1 − 5x3 = −3
4x1 − 5x2 + 3x3 = 3 (x1 = −8/7 , x2 = −10/7 , x3 = 1/7)
−2x1 + x2 + x3 = 1

3x1 + 4x2 − 3x3 = 5

3x1 − 2x2 + 4x3 = 7 (x1 = −7/3 , x2 = 9 , x3 = 8)
3x1 + 2x2 − x3 = 3

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Math210E Linear Systems and Matrices 42

Use Cramer’s rule to find y.

x + 2y + w = 2
2x − 3y + 3w = 1
(y = 1/10)
−x + 4y + 4z − w = −1
y − z = 0

x + 2y + z + w = 8
−x − y + 2z = 3
(y = 2)
2x + 3y = 0
−2x + y − 2z − w = 0

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Math210E Linear Systems and Matrices 43

 
1 4 1 h iT h iT
Let A =  2 7 4  , b = 1 1 2 , x = x1 x2 x3 .
 

2 8 3
Show that A is an invertible matrix. (|A| = −1 6= 0)

 
 11 4 −9 
Find A−1 by using elementary row operations. 
 −2 −1 2 

−2 0 1

For the linear equation system Ax = b, find x2 using Cramer’s rule. (x2 = 1)
Math210E Linear Systems and Matrices 44

Find A−1 for the following matrices A by using the adjoint matrix.
   
−5 −2 2 4 4 4 
1

A= 1

5 −3   16 15 13 

4

5 −3 1 28 25 23

   
2 4 −3 9 12 −13 
1 

A =  2 −3 −1   11 −21 −4 
 
107

−5 0 −3 −15 −20 −14

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Math210E Linear Systems and Matrices 45

 
3 0 1
Let A =  −2 1 0 
.


0 1 −2
 
 −2 1 −1 
Find the adjoint matrix of A.

 −4 −6 −2 

−2 −3 3

 
1/4 −1/8 1/8
Find A−1 . 
 1/2 3/4 1/4 

1/4 3/8 −3/8

2017 Spring prepared by Ayşe Peker Dobie

Math210E Linear Systems and Matrices 46

 
2 2 2
Let A =  1 x x2 . Find |A| and the values of x and y for which A is invertible.
 

1 y y2
(2(x − 1)(y − 1)(y − x) , x 6= 1 , y 6= 1 , x 6= y)

1 1 1 0
 
 1 0 0 1 
Show that the homogeneous system Ax = 0 has only the trivial solution for A =  .
 
 0 2 0 0 
1 0 1 1
(|A| = 2 6= 0)

2017 Spring prepared by Ayşe Peker Dobie

Math210E Linear Systems and Matrices 47

Prove the following properties.

   

 ka11 a12 a13 


 a11 a12 a13 


 ka21 a22 a23 
 =k

 a21 a22 a23 

   
 ka31 a32 a33   a31 a32 a33 

   

 a11 + ka12 a12 a13 


a11 a12 a13 


 a21 + ka22 a22 a23 
 = a21 a22 a23




   
 a31 + ka32 a32 a33  a31 a32 a33 

2017 Spring prepared by Ayşe Peker Dobie

Math210E Linear Systems and Matrices 48

Let A = [aij ] be a 3 × 3 matrix. Show that |AT | = |A| by expanding |A| along its first row and
|AT | along its first column.

Suppose that A2 = A. Prove that |A| = 0 or |A| = 1.

Suppose that An = 0 for some positive integer n. Prove that |A| = 0.

2017 Spring prepared by Ayşe Peker Dobie

Math210E Linear Systems and Matrices 49

The square matrix A is called orthogonal provided that AT = A−1 . Show that the determinant
of such a matrix must be either 1 or −1.

The matrices A and B are said to be similar provided that A = P −1 BP for some invertible matrix
P . Show that if A and B are similar, then |A| = |B|.

Suppose that AAT = A−1 B. Express |B| in terms of |A|.

2017 Spring prepared by Ayşe Peker Dobie