Worksheets1. Matrix, Rank of Matrix + Inverse + SLEs
Total questions: 143
Worksheet time: 2hrs 52mins
Let A = [1 2 3 -4], B = [5 0 -6 7]. Find 5A - 2B.
[-5 10 27 -34]
[-5 10 3 -6]
[5 10 15 -20]
[10 0 -12 14]
Undefined matrix operation
Let A = [1 2 3 -4], B = [5 0 -6 7]. Find 2A - B.
[-3 4 12 -15]
[2 4 6 -8]
[5 0 -6 7]
[10 0 -12 -14]
Undefined matrix operation
Let A = [1 2 3 -4], B = [5 0 -6 7]. Find A + 4B.
[21 2 -21 24]
[21 2 3 -4]
[20 0 -24 28]
[20 0 12 14]
Undefined matrix operation
Let A = [1 2 3 -4], B = [5 0 0 -6 7 0]. Find 3B - A.
[14 -2 -21 25]
[15 0 -18 21]
[1 2 0 3 -4 0]
[15 0 -18 21]
Undefined matrix operation
Let A = [1 2 3 -4], B = [5 0 1 -6 7 0]. Find AB.
[-7 14 1 39 -28 3]
[5 0 1 -12 14 0]
[-13 34 21 -28 1 2]
[15 0 -18 21]
Undefined matrix operation
Let A = [1 2 3 -4], B = [5 0 1 -6 7 0]. Find BA.
[-7 14 1 39 -28 3]
[5 0 1 -12 14 0]
[1 0 3 7]
[-13 34 21 -28 1 2]
Undefined matrix operation
Let A = [1 2 3 -4], B = [5 0 1 -6 7 0]. Find B^T A.
[-13 34 21 -28 1 2]
[5 0 1 -12 14 0]
[5 10 21 -28 1 2]
[15 0 -18 21]
Undefined matrix operation
Let A = [1 2 3 -4]. Find A^2.
[7 -6 -9 22]
[1 4 9 16]
[5 10 21 -28 1 2]
[1 0 18 21]
Undefined matrix operation
Let A = [1 2 3 -4]. Find A^3.
[-11 38 57 -106]
[1 4 9 16]
[-11 8 27 64]
[1 0 18 21]
Undefined matrix operation
Let A = [1 0 -2 3 10 0]. Find A^2.
[1 0 5 9 11 -4]
[1 4 9 16]
[1 0 4 9 100 0]
[1 0 18 21]
Undefined matrix operation
Calculate determinate |3 -2 4 0|.
0
10
8
-18
-16
Calculate determinate |2 0 6 -10|.
0
-2
-20
-14
-10
Calculate determinate |3 -2 1 4|.
10
12
14
-14
-10
Calculate determinate |√a 0 a √a|.
0
√a
a
2a
4√a
Calculate determinate |2 7 -1 6|.
0
-7
19
6
13
Calculate determinate |0 7 1 6|.
-7
0
7
6
-13
Calculate determinate |3 -2 4 6|.
0
10
26
-18
-16
Calculate determinate |2 3 6 -10|.
0
-2
-38
-14
-10
Calculate determinate |3 − 2 0 4 |
10
-14
12
14
-10
Calculate determinate |√a -1; a √a|
0
√a
2a
a
4√a
Calculate determinate |2 3 4 5 − 2 1 1 2 3 |
74
72
-10
-4
6
Calculate determinate |−𝑎 1 𝑎 0 − 𝑎 − 1 𝑎 1 −𝑎 |
2a
-2a/3
-2a
2a/3
3a/2
Calculate determinate |12 6 -4; 0 4 4; 0 0 8|
48
380
384
16
-16
Calculate determinate |-3 6 0; 0 4 4; 0 2 8|
48
88
-72
-18
-16
Calculate determinate [[-3, -6, -3], [4, 8, 4], [4, 8, 4]]
48
92
0
-42
-16
Let A = [1 8 0 9 -7 1], B = [-1 -7 0 1 6 -1]. Find A - 3B.
[4 27 0 6 -20 4]
[4 29 0 6 -25 4]
Undefined matrix operation
[4 0 -25 29 6 4]
[8 54 0 12 -40 8]
Let A = [2 -3 1 -1 0 -2], B = [4 3 2 -3 1 -4]. Find 5A - 2B.
[4 -21 10 2 -2 2]
[2 20 0 5 -21 7]
[2 -11 1 0 -2 -4]
[2 -21 1 1 -2 -2]
Undefined matrix operation
Let A = [1 -2 4 2 0 -1], B = [5 2 3 4 6 2]. Find A + B.
[6 0 7 6 6 1]
[6 6 0 6 7 1]
[6 0 7 6 -6 4]
[3 6 11 3 5 -7]
Undefined matrix operation
Let A = [3 5 5 8], B = [4 8 3 0 1 8]. Find 3A - B.
[5 7 15 23 -3 4]
[5 27 0 -6 5 4]
Undefined matrix operation
[5 15 -3 7 23 8]
[1 4 0 2 4 8]
Let A = [2 2 1 2], B = [3 4 2 3]. Find AB.
[10 7 14 10]
[10 10 14 7]
[10 14 7 10]
Undefined matrix operation
[1 7 4 10]
Let A = [2 0 -1 0 -2 2], B = [4 1 0 3 2 1 0 1 0]. Find AB.
[8 1 0 -6 -2 2]
[8 10 0 6 2 2]
[8 1 0 -6 -2 2 8 6 4]
[8 -6 10 1 -2 1 0 2 5]
Undefined matrix operation
Calculate determinate |8 7 0 9 |
62
65
0
-72
72
Calculate determinate
7
17
-3
3
0
Calculate determinate
32
-32
-18
-4
4
Calculate determinate
23
20
5
-17
-20
Calculate determinate
-4
14
6
-6
11
35. Calculate determinate
32
-32
-18
-4
4
36. Calculate determinate
23
20
5
-17
-20
37. Find elements of Matrix A.
-1
1
2
38. Find correct answer for y - 5 - 4y - 6 + 2xy + x + 4x + 6yy
y - 2x + 5yx - 4y4x + 67y
y + 2x + 5yx - 4y4x + 67y
-y + 2x + 5yx + 4y4x + 65y
-y + 2x + 5yx + 4y4x + 67y
39. For the matrices A = [[1, 2, 2], [3, 0, 2], [0, 2, 2]] and B = [[1, 5, 3], [0, 2, 1], [2, 3, 0]], find element c11 of the product C = A·B
5
-14
-7
1
-5
40. For the matrices A = [[1, 2, 2], [3, 0, 2], [0, 2, 2]] and B = [[1, 5, 3], [0, 2, 1], [2, 3, 0]], find element c12 of the product C = A·B
-7
-14
7
1
-5
41. For the matrices A = [[1, 2, 2], [3, 0, 2], [0, 2, 2]] and B = [[1, 5, 3], [0, 2, 1], [2, 3, 0]], find element c13 of the product C = A·B
1
-14
7
-7
-5
42. For the matrices A = [[1, 2, 2], [3, 0, 2], [0, 2, 2]] and B = [[1, 5, 3], [0, 2, 1], [2, 3, 0]], find element c21 of the product C = A·B
-1
-14
7
-7
-5
43. For the matrices A = [[1, 2, 2], [3, 0, 2], [0, 2, 2]] and B = [[1, 5, 3], [0, 2, 1], [2, 3, 0]], find element c22 of the product C = A·B
-9
-14
2
-7
-5
44. For the matrices A = [[1, 2, 2], [3, 0, 2], [0, 2, 2]] and B = [[1, 5, 3], [0, 2, 1], [2, 3, 0]], find element c23 of the product C = A·B
9
-14
2
-7
-5
45. For the matrices A = [[1, 2, 2], [3, 0, 2], [0, 2, 2]] and B = [[1, 5, 3], [0, 2, 1], [2, 3, 0]], find element c31 of the product C = A·B
-4
-14
2
-7
-5
46. For the matrices A = [[1, 2, 2], [3, 0, 2], [0, 2, 2]] and B = [[1, 5, 3], [0, 2, 1], [2, 3, 0]], find element c32 of the product C = A·B
2
-4
10
-7
-5
Find element c32 of the product С = A·B
-4
-14
7
2
-5
Find element c31 of the product С = B·A
-7
-14
7
-4
-5
Find element c23 of the product С = B·A
2
9
7
-7
-5
Find element c21 of the product С = B·A
-6
-1
7
10
-5
Find element c13 of the product С = B·A
6
-14
1
-7
-5
Find element c12 of the product С = B·A
4
-14
7
-7
-5
A square matrix is called diagonal, if
elements that do not lie on the main diagonal are zero
elements, lying on the second diagonal, are equal to zero
elements, lying on the main diagonal, are zero
elements, lying below the main diagonal, are zero
elements, lying upper the main diagonal, are equal to zero
A square matrix is called upper triangular, if
elements, lying below the main diagonal, are zero
elements, lying on the second diagonal, are equal to zero
elements, lying on the main diagonal, are zero
elements that do not lie on the main diagonal are zero
elements, lying upper the main diagonal, are equal to zero
A square matrix is called lower triangular, if
elements, lying upper the main diagonal, are zero
elements, lying on the second diagonal, are equal to zero
elements, lying on the main diagonal, are zero
elements that do not lie on the main diagonal are zero
elements, lying below the main diagonal, are equal to zero
Row operation over a matrix is
Row operation over a matrix is
interchange of two rows or two columns
transpose of rows and columns
finding its determinant
finding its rank
finding its basis
Row operation over a matrix is
multiplication of its column or row by a nonzero number
transpose of its rows and columns
finding its determinant
finding its rank
finding its basis
Row operation over a matrix is
addition of one row, multiplied by a number, to another row
transpose of its rows and columns
finding its determinant
finding its rank
finding its basis
Solve the equation 0 = x + x + x
21/4
4/21
1
5
0
Find determinant of the matrix B, where B=A•A^t
21
0
1
2
12
If matrices A and B, then determinant of the matrix A·B is equal to:
-32
0
1
32
10
Distribute the matrices in increasing order of their determinants:
B, A, C, D, E
C, B, A, D, E
C, D, E, B, A
D, E, B, A, C
C, A, B, D, E
If a row of a square matrix is replaced by sum of this row to another row, multiplied by a number a, then its determinant
is not changed
changes its sign
is multiplied to a
is equal to zero
is doubled
If we change places of two rows (two columns) of a square matrix, then its determinant
changes its sign
is not changed
is multiplied to a
is equal to zero
is doubled
If we multiply one row of a square matrix by a c ≠ 0, then its determinant
is multiplied to c
is not changed
changes its sign
is equal to zero
is doubled
If we multiply one row of a square matrix by 2, then its determinant
is doubled
is not changed
changes its sign
is equal to zero
is multiplied to c
If we multiply one column of a square matrix by 2, then its determinant
is doubled
is not changed
changes its sign
If we multiply one column of a square matrix by 2, then its determinant
is doubled
is not changed
changes its sign
is equal to zero
is multiplied to c
Find rank of the following matrices (write only number)
Find rank of the following matrices (write only number)
Find rank of the following matrices (write only number)
Find rank of the following matrices (write only number)
Find rank of the following matrices (write only number)
Find rank of the following matrices (write only number)
Find rank of the following matrices (write only number)
Find the inverse matrix A to the matrix
1. Find the inverse matrix -1 A to the matrix
5 9
7 4
2. Find the inverse matrix -1 A to the matrix
1 2
7 4
3. Find the inverse matrix -1 A to the matrix
1 8
6 10
4. Find the inverse matrix -1 A to the matrix
1 3
7 4
5. Find the inverse matrix -1 A to the matrix
4 1
5 3
6. Find the inverse matrix -1 A to the matrix
6 4
2 1
Find the rank of matrix |−3 6 0 0 4 4 1 2 8|
0
2
3
1
-1
Find the rank of matrix |2 8 20 0 4 4 1 2 8|
0
2
3
1
-1
Find the rank of matrix |−2 8 20 0 − 4 4 1 0 8|
0
3
2
1
-1
Find the rank of matrix |−2 8 20 1 − 4 −10 4 − 16 − 40|
0
1
2
3
-1
Find the rank of matrix |−2 8 20 1 0 −10 4 − 16 3|
0
3
2
1
-1
Let A = [1 2 3 − 4]. Find the inverse matrix A−1 =?
[4 10 2 10 3 10 −1 10]
[4 10 2 10 −3 10 1 10]
[1 10 2 10 −3 10 4 10]
[1 10 −2 10 −3 10 4 10]
[4 10 −2 10 −3 10 −1 10]
Let A = [4 − 1 2 5]. Find the inverse matrix A−1 =?
[5 22 1 22 −1 11 2 11]
[-5 22 1 22 −1 11 4 22]
[5 22 1 11 −2 11 2 11]
[1 11 −2 11 −3 11 2 11]
[4 11 −2 11 3 11 −1 22]
Let A = [3 10 1 5 2 5 3 5]. Find the inverse matrix A−1 =?
[6 − 2 − 4 3]
[3 − 2 − 4 6]
[6 4 8 3]
[6 − 2 4 − 3]
[3 2 − 4 − 6]
Let A = [3 7 − 2 7 − 4 7 5 7]. Find the inverse matrix A−1 =?
[5 2 4 3]
[5 − 2 − 4 3]
[3 − 2 − 4 5]
[3 2 4 5]
[5 − 2 4 3]
Let A = [1 3 2 3 0 1]. Find the inverse matrix A−1 =?
[3 − 2 0 1]
[3 2 0 1]
[1 − 2 0 3]
[3 0 2 1]
[1 0 3 2]
Point out matrices, which have inverses
B and C
B, C, D, E
A
B, D and C
A and C
Use the invertible matrix property to determine the value(s) of a for which the matrix |−−−−| is invertible.
a ≠ 0, a ≠ -2, a ≠ 2
a ≠ 0, a ≠ -2
a ≠ 0, a ≠ -1, a ≠ 1
a ≠ 0
a is any real number
Use the invertible matrix property to determine the value(s) of a for which the matrix |100 111 01 a| is invertible.
a ≠ 1
a ≠ 0, a ≠ -2
a ≠ 0, a ≠ -1, a ≠ 1
a ≠ 0
a is any real number
Find the value(s) of a for which the matrix 100 111 01 a is invertible. ● a ≠ 1 o a ≠ 0, a ≠ -2 o a ≠ 0, a ≠ -1, a ≠ 1 o a ≠ 0 o a is any real number
a ≠ 1
a ≠ 0, a ≠ -2
a ≠ 0, a ≠ -1, a ≠ 1
a ≠ 0
a is any real number
Use the invertible matrix property to determine the value(s) of b for which the matrix 112 123 01 a is NOT invertible. ● a = 1 o a = 0, a = -2 o a = 0, a = -1, a = 1 o a ≠ 0 o a is any real number
a = 1
a = 0, a = -2
a = 0, a = -1, a = 1
a ≠ 0
a is any real number
Arrange the matrices in descending order of their ranks: A = 5 5 43 43 21 21; B = 600 540 321; C = 000 000 000; D = 5 5 43 43 21 21 ● B, D, A, C o C, B, A, D o C, D, B, A o D, B, A, C o C, A, B, D
B, D, A, C
C, B, A, D
C, D, B, A
D, B, A, C
C, A, B, D
Point out the correct statements, relating to definition and existence of an inverse matrix: ● inverse matrix A -1 exists, if the matrix A is a square and det A ≠ 0 o A·A -1 = A -1 ·A = A o A·A -1 = A -1 ·A = 1 o inverse matrix A -1 exists, if the matrix A is a square o inverse matrix A -1 exists, if rank(A) = 1
inverse matrix A -1 exists, if the matrix A is a square and det A ≠ 0
A·A -1 = A -1 ·A = A
A·A -1 = A -1 ·A = 1
inverse matrix A -1 exists, if the matrix A is a square
inverse matrix A -1 exists, if rank(A) = 1
If A·x = b, then ● x = A -1 ·b o x = b· A -1 o x = b/A o x = b· A o x = A·b
x = A -1 ·b
x = b· A -1
x = b/A
x = b· A
x = A·b
Choose the correct statement. Rank of the matrix is equal to... ● number of non-zero rows in a reduced matrix o number of columns o product of the number of rows to number of columns o the number of rows in the matrix o its determinant
number of non-zero rows in a reduced matrix
number of columns
product of the number of rows to number of columns
the number of rows in the matrix
its determinant
Find the rank of matrix | 1 0 1 0 1 0 0 3 2 |
0
3
2
1
-1
Find the rank of matrix | 1 5 1 2 4 0 4 8 0 |
0
3
2
1
-1
Find the rank of matrix | 2 8 10 4 1 7 -1 -5 10 |
0
3
2
1
-1
Let A = [5 4 0 -2]. Find the inverse matrix A -1 =?
[1 5 2 5 -3 10 1 10]
[4 10 2 10 3 10 -1 10]
[1 5 2 5 0 -1 2]
[1 10 -2 10 -3 10 4 10]
[4 10 -2 10 0 -1 10]
Let A = [-3 4 2 6]. Find the inverse matrix A -1 =?
[-3 13 2 13 1 13 3 26]
[-3 23 2 13 1 23 3 26]
[0 2 13 1 16 3 26]
[3 13 -2 13 -1 13 3 26]
[6 13 2 13 1 13 3 26]
Let A = [4 3 -2 -7]. Find the inverse matrix A -1 =?
[-7 22 3 22 -1 11 -2 11]
[7 22 -3 42 -1 11 -2 11]
[7 22 3 22 -21 11 -2 11]
[7 22 3 22 -1 21 -2 41]
[7 22 3 22 -1 11 -2 11]
Find the rank of matrix: | 1 -3 2 -4 3 3 1 0 |
Find the rank of matrix: 1 - 3 2 4 3 3 1 0 1
0
1
2
3
Find the rank of matrix: 8 5 11 11 7 15 5 3 7 2 1 3
4
2
1
3
Find the rank of matrix: 0 0 1 1 1 1 1 0 1 1 2 1
4
2
1
3
In solving systems of equations by Cramer's rule we use formulas:
Δi/Δ = xi
Δ·Δ = ii
xi/Δ = i
Δ - Δ = ii
A solution of a system of equations is an ordered set of real numbers
which inverts each equation of the system in a true numeric identity
which all are zero
which inverts determinant to zero
which is equal to a number of rows
which is equal to rank of augmented matrix
A system of equations is called compatible (consistent) if
it has at least one solution
it has no solution
determinant is equal to 0
rank of the coefficients matrix is more than rank of augmented matrix
number of rows is equal to the number of columns
A system of equations is called non-compatible (inconsistent) if
it has no solution
it has at least one solution
determinant is equal to 0
rank of the coefficients matrix is equal to rank of augmented matrix
number of rows is equal to the number of columns
A system of equations is called determinate if
it is compatible and has unique solution
it has at least one solution
it has no solution
it has infinitely many solutions
number of rows is equal to the number of columns
A system of equations is called indeterminate if
it is compatible and has infinitely many solutions
it has at least one solution
it has no solution
it is compatible and has unique solution
number of rows is equal to the number of columns
Two systems of linear equations are equivalent if
they have the same solution set
their ranks are equal
they have the same numbers of rows and columns
their dimensions are equal
their determinants are equal
A system of linear equations of the form AX=0 is called
homogenous system
consistent system
equivalent system
A system of linear equations of the form AX=0 is called
homogenous system
consistent system
equivalent system
reduced system
zero system
Solve the linear system
x = 3, y = 2
x = 5, y = 2
x = 5, y = 4
x = 6, y = 11
x = 11, y = 2
Solve the linear system
x = 3, y = 21
x = 5, y = 12
x = 5, y = 21
x = 7, y = 21
x = 3, y = 21
Solve the linear system
x = 5, y = 21
x = 7, y = 21
x = 5, y = 12
x = 7, y = 25
x = 5, y = 2
Solve the linear system
x = 7, y = 21
x = 5, y = 12
x = 7, y = 34
x = 7, y = 19
x = 7, y = 21
Solve the linear system
x = 5, y = 21
x = 5, y = 12
x = 25, y = 32
x = 5, y = 18
x = 5, y = 12
Solve the linear system
x = 25, y = 1
x = 27, y = 1
x = 21, y = 1
x = 25, y = 12
x = 25, y = 12
Solve the linear system
x = 13, y = 1
x = 13, y = 1
x = 13, y = 12
x = 13, y = 11
x = 13, y = 12
Solve the linear system
x = 17, y = 1
x = 17, y = 3
x = 17, y = 6
x = 17, y = 1
x = 17, y = 14
Solve the linear system
x = 7, y = 22
x = 17, y = 22
x = 7, y = 22
x = 7, y = 2
x = 7, y = 22
Solve the system of linear equations
x = 1, y = 2, z = 5
x = 1, y = 2, z = 5
x = 1, y = 5, z = 2
x = 5, y = 2, z = 1
x = 1, y = 4, z = 2
Solve the system of linear equations
x = 1, y = 3, z = 2
x = 2, y = 2, z = 5
x = 3, y = 2, z = 2
x = 1, y = 2, z = 5
x = 2, y = 5, z = 3
Solve the system of linear equations 132 225 322
Solve the system of linear equations 132 252 322
Solve the system of linear equations 12 3523 323
Solve the system of linear equations 12 532 222
Solve the system of linear equations 3421 3429 210
Solve the system of linear equations 3 2 5 5 2 3 4 12 2 3 1
3;2;1;x;y;z
2;3;0;x;y;z
2;4;3;x;y;z
3;2;0;x;y;z
3;5;0;x;y;z
Solve the system of linear equations 4 4 19 2 2 11 2 8
1;2;4;x;y;z
4;1;1;x;y;z
1;5;2
5;2;1
1;1;4;xyz
Solve the system of linear equations 2 2 0 4 4 6 2 4
0;2;1;x;y;z
2;4;0;xyz
0;2;1;x;y;z
2;6;3;xyz
4;0;2;xyz
Solve the system of linear equations 228 211 4422
