Algebra 2 â Matrix Algebra Tutor - Worksheet 1
Algebra 2 – Matrix Algebra Tutor Worksheet 1 Introduction to Matrices
1 © MathTutorDVD.com
Algebra 2 – Matrix Algebra Tutor - Worksheet 1 – Introduction to Matrices 1. State the order of this matrix. −4 [ −2
9 2 8 ] 0 2 3
2. State the order of this matrix. Then identify the elements of the matrix that have a value of 1. 1 4 6 [2 −2 1 6 −8 1
1 4] −1
3. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = 2, 𝑎21 = 6, 𝑎13 = 1, 𝑎22 = 5, 𝑎23 = 3, 𝑎12 = 4
2 © MathTutorDVD.com
4. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = −4, 𝑎21 = 0, 𝑎13 = 17, 𝑎12 = −9, 𝑎32 = −2, 𝑎24 = 6, 𝑎34 = 0, 𝑎31 = −3, 𝑎14 = 1, 𝑎23 = 8, 𝑎33 = 10, 𝑎22 = 5
5. Find the value of the variables that make the matrix equation true. [
3𝑥 8
9 −4 ]=[ 5𝑤 2𝑧
4𝑦 ] −25
6. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = 3, 𝑎21 = 0, 𝑎13 = 6, 𝑎12 = 12, 𝑎32 = −7, 𝑎31 = −8, 𝑎23 = 9, 𝑎33 = 4, 𝑎22 = −6
3 © MathTutorDVD.com
7. Are these two matrices equal? Justify your answer. 3 −1 7 −2 −9 7 [2 6 −9] , [ 4 6 −1] −5 4 −2 −5 2 3
8. Are these two matrices equal? Justify your answer. −22 [−22 36 −48], [ 36 ] −48
9. Find the value of the variables that make the matrix equation true. 2𝑎 + 4 8𝑑 − 6 [ −7𝑐 − 1 9𝑒 − 6
6𝑏 + 2 10 18 −34 ] ]=[ 5𝑓 + 1 −36 30 26
4 © MathTutorDVD.com
10. Find the value of the variables that make the matrix equation true. −4𝑞 8 2𝑟 − 6 [4𝑠 + 1 22 ] = [ 17 −17 11 −3𝑢 + 4
12 9𝑡 + 4 ] 2𝑣 − 1
11. Find the value of the variables that make the matrix equation true. 2𝑥 24 −63 40 [−27 36 −72] = [3𝑦 4𝑧 32 54 48
5 © MathTutorDVD.com
7𝑦 3𝑥 −6𝑦
5𝑧 −9𝑧] 4𝑥
12. Find the value of the variables that make the matrix equation true. 8 [0 0
7𝑥 + 3 0 0 0 0 9𝑦 − 4 0 ] 8 0] = [ 0 0 8 0 0 5𝑧 − 5
6 © MathTutorDVD.com
Answers - Algebra 2 – Matrix Algebra Tutor - Worksheet 1 – Introduction to Matrices 1. State the order of this matrix. −4 [ −2
9 2 8 ] 0 2 3
The order of a matrix describes how many rows and how many columns a matrix contains. The given matrix has 2 rows and 4 columns, so its order is 2 by 4. Answer: 2 by 4
2. State the order of this matrix. Then identify the elements of the matrix that have a value of 1. 1 4 6 [2 −2 1 6 −8 1
1 4] −1
The order of a matrix describes how many rows and how many columns a matrix contains. The given matrix has 3 rows and 4 columns, so its order is 3 by 4. Answer: 3 by 4
3. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = 2, 𝑎21 = 6, 𝑎13 = 1, 𝑎22 = 5, 𝑎23 = 3, 𝑎12 = 4 The elements in matrices are arranged in the following manner. 𝑎11 [ ⋮ 𝑎𝑛1
⋯ 𝑎1𝑛 ⋱ ⋮ ] ⋯ 𝑎𝑛𝑛
Where 𝑎𝑖𝑗 denotes the element in row 𝑖 and column 𝑗. 7 © MathTutorDVD.com
The given elements are 𝑎11 , 𝑎12 , 𝑎13 , 𝑎21 , 𝑎22 , 𝑎23 which means the matrix has two rows and two columns, a 2 by 3 matrix, arranged in the following manner. 𝑎11 [𝑎
21
𝑎12 𝑎22
𝑎13 𝑎23 ]
Therefore, the given elements form the matrix: 2 [ 6 Answer: [
4 1 ] 5 3
2 4 1 ], 2 by 3 6 5 3
4. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = −4, 𝑎21 = 0, 𝑎13 = 17, 𝑎12 = −9, 𝑎32 = −2, 𝑎24 = 6, 𝑎34 = 0, 𝑎31 = −3, 𝑎14 = 1, 𝑎23 = 8, 𝑎33 = 10, 𝑎22 = 5 The elements in matrices are arranged in the following manner. 𝑎11 [ ⋮ 𝑎𝑛1
⋯ 𝑎1𝑛 ⋱ ⋮ ] ⋯ 𝑎𝑛𝑛
Where 𝑎𝑖𝑗 denotes the element in row 𝑖 and column 𝑗. The given elements are 𝑎11 , 𝑎12 , 𝑎13 , 𝑎14 , 𝑎21 , 𝑎22 , 𝑎23, 𝑎24 , 𝑎31 , 𝑎32 , 𝑎33 , 𝑎34 which means the matrix has three rows and four columns, which is a 3 by 4 matrix, arranged in the following manner. 𝑎11 [𝑎21 𝑎31
𝑎12 𝑎13 𝑎22 𝑎23 𝑎32 𝑎33
𝑎14 𝑎24 ] 𝑎34
Therefore, the given elements form the matrix: −4 [0 −3
−9 17 1 5 8 6] −2 10 0 8
© MathTutorDVD.com
−4 −9 17 1 Answer: [ 0 5 8 6], 3 by 4 −3 −2 10 0
5. Find the value of the variables that make the matrix equation true. [
3𝑥 8
9 −4 ]=[ 5𝑤 2𝑧
4𝑦 ] −25
In order for the matrix equation to be true, the elements in one matrix must be equal to the corresponding elements in the other matrix. Therefore, 3𝑥 = 9, 𝑥 = 3; −4 = 4𝑦, 𝑦 = −1; 8 = 2𝑧, 𝑧 = 4, and 5𝑤 = −25, 𝑤 = −5. Answer: 𝑥 = 3; 𝑦 = −1; 𝑧 = 4, and 𝑤 = −5
6. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = 3, 𝑎21 = 0, 𝑎13 = 6, 𝑎12 = 12, 𝑎32 = −7, 𝑎31 = −8, 𝑎23 = 9, 𝑎33 = 4, 𝑎22 = −6 The elements in matrices are arranged in the following manner. 𝑎11 [ ⋮ 𝑎𝑛1
⋯ 𝑎1𝑛 ⋱ ⋮ ] ⋯ 𝑎𝑛𝑛
Where 𝑎𝑖𝑗 denotes the element in row 𝑖 and column 𝑗. The given elements are 𝑎11 , 𝑎12 , 𝑎13 , 𝑎21 , 𝑎22 , 𝑎23, 𝑎31 , 𝑎32 , 𝑎33 which means the matrix has three rows and three columns, which is a 3 by 3 matrix, arranged in the following manner.
9 © MathTutorDVD.com
𝑎11 [𝑎21 𝑎31
𝑎12 𝑎13 𝑎22 𝑎23 ] 𝑎32 𝑎33
Therefore, the given elements form the matrix: 3 [0 −8
12 6 −6 9] −7 4
3 12 6 Answer: [ 0 −6 9], 3 by 3 −8 −7 4
7. Are these two matrices equal? Justify your answer. 3 −1 7 −2 −9 7 [2 6 −9] , [ 4 6 −1] −5 4 −2 −5 2 3 In order for the matrices to be equal the matrices must have the same order and the elements in one matrix must be equal to the corresponding elements in the other matrix. The two given matrices have the same order and the same element values, but the matching values are not in corresponding positions. Therefore, the matrices are not equal. Answer: The matrices are not equal because corresponding elements are not equal to each other.
8. Are these two matrices equal? Justify your answer. −22 [−22 36 −48], [ 36 ] −48 10 © MathTutorDVD.com
In order for the matrices to be equal the matrices must have the same order and the elements in one matrix must be equal to the corresponding elements in the other matrix. The two given matrices have the same element values, but the matrices do not have the same order. The first matrix is a 1 by 3 matrix and the second matrix is a 3 by 1 matrix. Therefore, the matrices are not equal. Answer: The matrices are not equal because the matrices do not have the same order.
9. Find the value of the variables that make the matrix equation true. 2𝑎 + 4 8𝑑 − 6 [ −7𝑐 − 1 9𝑒 − 6
6𝑏 + 2 10 18 −34 ] ]=[ 5𝑓 + 1 −36 30 26
In order for the matrix equation to be true, the elements in one matrix must be equal to the corresponding elements in the other matrix. Therefore, 2𝑎 + 4 = 10, 𝑎 = 3; 8𝑑 − 6 = 18, 𝑑 = 3; 6𝑏 + 2 = −34, 𝑏 = −6; −7𝑐 − 1 = −36, 𝑐 = 5; 9𝑒 − 6 = 30, 𝑒 = 4; and 5𝑓 + 1 = 26, 𝑓 = 5. Answer: 𝑎 = 3; 𝑑 = 3; 𝑏 = −6; 𝑐 = 5; 𝑒 = 4; and 𝑓 = 5
10. Find the value of the variables that make the matrix equation true. −4𝑞 8 2𝑟 − 6 [4𝑠 + 1 22 ] = [ 17 −17 11 −3𝑢 + 4
12 9𝑡 + 4 ] 2𝑣 − 1
In order for the matrix equation to be true, the elements in one matrix must be equal to the corresponding elements in the other matrix. Therefore, 8 = −4𝑞, 𝑞 = −2; 2𝑟 − 6 = 12, 𝑟 = 9; 4𝑠 + 1 = 17, 𝑠 = 4; 22 = 9𝑡 + 4, 𝑡 = 2; −17 = −3𝑢 + 4, 𝑢 = 7; and 11 = 2𝑣 − 1, 𝑣 = 6. 11 © MathTutorDVD.com
Answer: 𝑞 = −2; 𝑟 = 9; 𝑠 = 4; 𝑡 = 2; 𝑢 = 7; and 𝑣 = 6
11. Find the value of the variables that make the matrix equation true. 2𝑥 24 −63 40 [−27 36 −72] = [3𝑦 4𝑧 32 54 48
7𝑦 3𝑥 −6𝑦
5𝑧 −9𝑧] 4𝑥
In order for the matrix equation to be true, the elements in one matrix must be equal to the corresponding elements in the other matrix. Therefore, 24 = 2𝑥, 𝑥 = 12; −63 = 7𝑦, 𝑦 = −9; and 40 = 5𝑧, 𝑧 = 8 Other calculations include −27 = 3𝑦, 𝑦 = −9; 36 = 3𝑥, 𝑥 = 12; and − 72 = −9𝑧; 𝑧 = 8 and 32 = 4𝑥, 𝑧 = 8; 54 = −6𝑦, 𝑦 = −9; and 48 = 4𝑥, 𝑥 = 12 Answer: 𝑥 = 12; 𝑦 = −9; and 𝑧 = 8
12. Find the value of the variables that make the matrix equation true. 8 [0 0
7𝑥 + 3 0 0 0 0 9𝑦 − 4 0 ] 8 0] = [ 0 0 8 0 0 5𝑧 − 5
In order for the matrix equation to be true, the elements in one matrix must be equal to the corresponding elements in the other matrix. 5
4
13
7
3
5
Therefore, 7𝑥 + 3 = 8, 𝑥 = ; 9𝑦 − 4 = 8, 𝑦 = ; and 5𝑧 − 5 = 8, 𝑧 = 5
4
13
7
3
5
Answer: 𝑥 = ; 𝑦 = ; and 𝑧 =
12 © MathTutorDVD.com
1 © MathTutorDVD.com
Algebra 2 – Matrix Algebra Tutor - Worksheet 1 – Introduction to Matrices 1. State the order of this matrix. −4 [ −2
9 2 8 ] 0 2 3
2. State the order of this matrix. Then identify the elements of the matrix that have a value of 1. 1 4 6 [2 −2 1 6 −8 1
1 4] −1
3. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = 2, 𝑎21 = 6, 𝑎13 = 1, 𝑎22 = 5, 𝑎23 = 3, 𝑎12 = 4
2 © MathTutorDVD.com
4. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = −4, 𝑎21 = 0, 𝑎13 = 17, 𝑎12 = −9, 𝑎32 = −2, 𝑎24 = 6, 𝑎34 = 0, 𝑎31 = −3, 𝑎14 = 1, 𝑎23 = 8, 𝑎33 = 10, 𝑎22 = 5
5. Find the value of the variables that make the matrix equation true. [
3𝑥 8
9 −4 ]=[ 5𝑤 2𝑧
4𝑦 ] −25
6. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = 3, 𝑎21 = 0, 𝑎13 = 6, 𝑎12 = 12, 𝑎32 = −7, 𝑎31 = −8, 𝑎23 = 9, 𝑎33 = 4, 𝑎22 = −6
3 © MathTutorDVD.com
7. Are these two matrices equal? Justify your answer. 3 −1 7 −2 −9 7 [2 6 −9] , [ 4 6 −1] −5 4 −2 −5 2 3
8. Are these two matrices equal? Justify your answer. −22 [−22 36 −48], [ 36 ] −48
9. Find the value of the variables that make the matrix equation true. 2𝑎 + 4 8𝑑 − 6 [ −7𝑐 − 1 9𝑒 − 6
6𝑏 + 2 10 18 −34 ] ]=[ 5𝑓 + 1 −36 30 26
4 © MathTutorDVD.com
10. Find the value of the variables that make the matrix equation true. −4𝑞 8 2𝑟 − 6 [4𝑠 + 1 22 ] = [ 17 −17 11 −3𝑢 + 4
12 9𝑡 + 4 ] 2𝑣 − 1
11. Find the value of the variables that make the matrix equation true. 2𝑥 24 −63 40 [−27 36 −72] = [3𝑦 4𝑧 32 54 48
5 © MathTutorDVD.com
7𝑦 3𝑥 −6𝑦
5𝑧 −9𝑧] 4𝑥
12. Find the value of the variables that make the matrix equation true. 8 [0 0
7𝑥 + 3 0 0 0 0 9𝑦 − 4 0 ] 8 0] = [ 0 0 8 0 0 5𝑧 − 5
6 © MathTutorDVD.com
Answers - Algebra 2 – Matrix Algebra Tutor - Worksheet 1 – Introduction to Matrices 1. State the order of this matrix. −4 [ −2
9 2 8 ] 0 2 3
The order of a matrix describes how many rows and how many columns a matrix contains. The given matrix has 2 rows and 4 columns, so its order is 2 by 4. Answer: 2 by 4
2. State the order of this matrix. Then identify the elements of the matrix that have a value of 1. 1 4 6 [2 −2 1 6 −8 1
1 4] −1
The order of a matrix describes how many rows and how many columns a matrix contains. The given matrix has 3 rows and 4 columns, so its order is 3 by 4. Answer: 3 by 4
3. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = 2, 𝑎21 = 6, 𝑎13 = 1, 𝑎22 = 5, 𝑎23 = 3, 𝑎12 = 4 The elements in matrices are arranged in the following manner. 𝑎11 [ ⋮ 𝑎𝑛1
⋯ 𝑎1𝑛 ⋱ ⋮ ] ⋯ 𝑎𝑛𝑛
Where 𝑎𝑖𝑗 denotes the element in row 𝑖 and column 𝑗. 7 © MathTutorDVD.com
The given elements are 𝑎11 , 𝑎12 , 𝑎13 , 𝑎21 , 𝑎22 , 𝑎23 which means the matrix has two rows and two columns, a 2 by 3 matrix, arranged in the following manner. 𝑎11 [𝑎
21
𝑎12 𝑎22
𝑎13 𝑎23 ]
Therefore, the given elements form the matrix: 2 [ 6 Answer: [
4 1 ] 5 3
2 4 1 ], 2 by 3 6 5 3
4. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = −4, 𝑎21 = 0, 𝑎13 = 17, 𝑎12 = −9, 𝑎32 = −2, 𝑎24 = 6, 𝑎34 = 0, 𝑎31 = −3, 𝑎14 = 1, 𝑎23 = 8, 𝑎33 = 10, 𝑎22 = 5 The elements in matrices are arranged in the following manner. 𝑎11 [ ⋮ 𝑎𝑛1
⋯ 𝑎1𝑛 ⋱ ⋮ ] ⋯ 𝑎𝑛𝑛
Where 𝑎𝑖𝑗 denotes the element in row 𝑖 and column 𝑗. The given elements are 𝑎11 , 𝑎12 , 𝑎13 , 𝑎14 , 𝑎21 , 𝑎22 , 𝑎23, 𝑎24 , 𝑎31 , 𝑎32 , 𝑎33 , 𝑎34 which means the matrix has three rows and four columns, which is a 3 by 4 matrix, arranged in the following manner. 𝑎11 [𝑎21 𝑎31
𝑎12 𝑎13 𝑎22 𝑎23 𝑎32 𝑎33
𝑎14 𝑎24 ] 𝑎34
Therefore, the given elements form the matrix: −4 [0 −3
−9 17 1 5 8 6] −2 10 0 8
© MathTutorDVD.com
−4 −9 17 1 Answer: [ 0 5 8 6], 3 by 4 −3 −2 10 0
5. Find the value of the variables that make the matrix equation true. [
3𝑥 8
9 −4 ]=[ 5𝑤 2𝑧
4𝑦 ] −25
In order for the matrix equation to be true, the elements in one matrix must be equal to the corresponding elements in the other matrix. Therefore, 3𝑥 = 9, 𝑥 = 3; −4 = 4𝑦, 𝑦 = −1; 8 = 2𝑧, 𝑧 = 4, and 5𝑤 = −25, 𝑤 = −5. Answer: 𝑥 = 3; 𝑦 = −1; 𝑧 = 4, and 𝑤 = −5
6. Create a matrix that has the following elements, and state the order of the matrix. 𝑎11 = 3, 𝑎21 = 0, 𝑎13 = 6, 𝑎12 = 12, 𝑎32 = −7, 𝑎31 = −8, 𝑎23 = 9, 𝑎33 = 4, 𝑎22 = −6 The elements in matrices are arranged in the following manner. 𝑎11 [ ⋮ 𝑎𝑛1
⋯ 𝑎1𝑛 ⋱ ⋮ ] ⋯ 𝑎𝑛𝑛
Where 𝑎𝑖𝑗 denotes the element in row 𝑖 and column 𝑗. The given elements are 𝑎11 , 𝑎12 , 𝑎13 , 𝑎21 , 𝑎22 , 𝑎23, 𝑎31 , 𝑎32 , 𝑎33 which means the matrix has three rows and three columns, which is a 3 by 3 matrix, arranged in the following manner.
9 © MathTutorDVD.com
𝑎11 [𝑎21 𝑎31
𝑎12 𝑎13 𝑎22 𝑎23 ] 𝑎32 𝑎33
Therefore, the given elements form the matrix: 3 [0 −8
12 6 −6 9] −7 4
3 12 6 Answer: [ 0 −6 9], 3 by 3 −8 −7 4
7. Are these two matrices equal? Justify your answer. 3 −1 7 −2 −9 7 [2 6 −9] , [ 4 6 −1] −5 4 −2 −5 2 3 In order for the matrices to be equal the matrices must have the same order and the elements in one matrix must be equal to the corresponding elements in the other matrix. The two given matrices have the same order and the same element values, but the matching values are not in corresponding positions. Therefore, the matrices are not equal. Answer: The matrices are not equal because corresponding elements are not equal to each other.
8. Are these two matrices equal? Justify your answer. −22 [−22 36 −48], [ 36 ] −48 10 © MathTutorDVD.com
In order for the matrices to be equal the matrices must have the same order and the elements in one matrix must be equal to the corresponding elements in the other matrix. The two given matrices have the same element values, but the matrices do not have the same order. The first matrix is a 1 by 3 matrix and the second matrix is a 3 by 1 matrix. Therefore, the matrices are not equal. Answer: The matrices are not equal because the matrices do not have the same order.
9. Find the value of the variables that make the matrix equation true. 2𝑎 + 4 8𝑑 − 6 [ −7𝑐 − 1 9𝑒 − 6
6𝑏 + 2 10 18 −34 ] ]=[ 5𝑓 + 1 −36 30 26
In order for the matrix equation to be true, the elements in one matrix must be equal to the corresponding elements in the other matrix. Therefore, 2𝑎 + 4 = 10, 𝑎 = 3; 8𝑑 − 6 = 18, 𝑑 = 3; 6𝑏 + 2 = −34, 𝑏 = −6; −7𝑐 − 1 = −36, 𝑐 = 5; 9𝑒 − 6 = 30, 𝑒 = 4; and 5𝑓 + 1 = 26, 𝑓 = 5. Answer: 𝑎 = 3; 𝑑 = 3; 𝑏 = −6; 𝑐 = 5; 𝑒 = 4; and 𝑓 = 5
10. Find the value of the variables that make the matrix equation true. −4𝑞 8 2𝑟 − 6 [4𝑠 + 1 22 ] = [ 17 −17 11 −3𝑢 + 4
12 9𝑡 + 4 ] 2𝑣 − 1
In order for the matrix equation to be true, the elements in one matrix must be equal to the corresponding elements in the other matrix. Therefore, 8 = −4𝑞, 𝑞 = −2; 2𝑟 − 6 = 12, 𝑟 = 9; 4𝑠 + 1 = 17, 𝑠 = 4; 22 = 9𝑡 + 4, 𝑡 = 2; −17 = −3𝑢 + 4, 𝑢 = 7; and 11 = 2𝑣 − 1, 𝑣 = 6. 11 © MathTutorDVD.com
Answer: 𝑞 = −2; 𝑟 = 9; 𝑠 = 4; 𝑡 = 2; 𝑢 = 7; and 𝑣 = 6
11. Find the value of the variables that make the matrix equation true. 2𝑥 24 −63 40 [−27 36 −72] = [3𝑦 4𝑧 32 54 48
7𝑦 3𝑥 −6𝑦
5𝑧 −9𝑧] 4𝑥
In order for the matrix equation to be true, the elements in one matrix must be equal to the corresponding elements in the other matrix. Therefore, 24 = 2𝑥, 𝑥 = 12; −63 = 7𝑦, 𝑦 = −9; and 40 = 5𝑧, 𝑧 = 8 Other calculations include −27 = 3𝑦, 𝑦 = −9; 36 = 3𝑥, 𝑥 = 12; and − 72 = −9𝑧; 𝑧 = 8 and 32 = 4𝑥, 𝑧 = 8; 54 = −6𝑦, 𝑦 = −9; and 48 = 4𝑥, 𝑥 = 12 Answer: 𝑥 = 12; 𝑦 = −9; and 𝑧 = 8
12. Find the value of the variables that make the matrix equation true. 8 [0 0
7𝑥 + 3 0 0 0 0 9𝑦 − 4 0 ] 8 0] = [ 0 0 8 0 0 5𝑧 − 5
In order for the matrix equation to be true, the elements in one matrix must be equal to the corresponding elements in the other matrix. 5
4
13
7
3
5
Therefore, 7𝑥 + 3 = 8, 𝑥 = ; 9𝑦 − 4 = 8, 𝑦 = ; and 5𝑧 − 5 = 8, 𝑧 = 5
4
13
7
3
5
Answer: 𝑥 = ; 𝑦 = ; and 𝑧 =
12 © MathTutorDVD.com
