Quiz1
Mathematics for Data Science II · Quiz 1 · May 2026
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Question 2 MCQ · 4.0 marks
Let [[IMAGE:4cd67eb32c1474f1_2_2]] be a [[IMAGE:4cd67eb32c1474f1_2_3]] matrix such that [[IMAGE:4cd67eb32c1474f1_2_4]] . Suppose [[IMAGE:4cd67eb32c1474f1_2_5]] is a matrix obtained from [[IMAGE:4cd67eb32c1474f1_2_6]] by
swapping the second and third rows, and then multiplying the first row by [[IMAGE:4cd67eb32c1474f1_2_7]] . Which of the
following options is correct?
[[IMAGE:4cd67eb32c1474f1_2_8]] .
[[IMAGE:4cd67eb32c1474f1_2_9]]
[[IMAGE:4cd67eb32c1474f1_2_10]]
[[IMAGE:4cd67eb32c1474f1_2_11]]
A published solution is not available for this question yet.
Question 3 MSQ · 4.0 marks
Let [[IMAGE:4cd67eb32c1474f1_2_12]] . Choose the correct options.
[[IMAGE:4cd67eb32c1474f1_3_13]] and rank [[IMAGE:4cd67eb32c1474f1_3_14]] .
[[IMAGE:4cd67eb32c1474f1_3_15]] and rank [[IMAGE:4cd67eb32c1474f1_3_16]] .
The matrix [[IMAGE:4cd67eb32c1474f1_3_17]] is in the reduced row echelon form (RREF).
The reduced row echelon form (RREF) of [[IMAGE:4cd67eb32c1474f1_3_18]] is [[IMAGE:4cd67eb32c1474f1_3_19]] .
rank [[IMAGE:4cd67eb32c1474f1_3_20]] rank [[IMAGE:4cd67eb32c1474f1_3_21]] .
A published solution is not available for this question yet.
Question 4 MSQ · 4.0 marks
Consider the vector space
[[IMAGE:4cd67eb32c1474f1_3_22]]
The subset
[[IMAGE:4cd67eb32c1474f1_3_23]]
is a linearly independent subset of [[IMAGE:4cd67eb32c1474f1_3_24]] . Which of the following matrices [[IMAGE:4cd67eb32c1474f1_3_25]] are vectors in [[IMAGE:4cd67eb32c1474f1_3_26]] such
that
[[IMAGE:4cd67eb32c1474f1_3_27]] is a basis for [[IMAGE:4cd67eb32c1474f1_3_28]] ?
[[IMAGE:4cd67eb32c1474f1_3_29]]
[[IMAGE:4cd67eb32c1474f1_3_30]]
[[IMAGE:4cd67eb32c1474f1_3_31]]
[[IMAGE:4cd67eb32c1474f1_3_32]]
[[IMAGE:4cd67eb32c1474f1_4_33]]
A published solution is not available for this question yet.
Question 5 NAT · 4.0 marks
If [[IMAGE:4cd67eb32c1474f1_4_34]] is a linear combination of the vectors [[IMAGE:4cd67eb32c1474f1_4_35]] as
[[IMAGE:4cd67eb32c1474f1_4_36]] , then find the value of [[IMAGE:4cd67eb32c1474f1_4_37]] .
A published solution is not available for this question yet.
Question 6 NAT · 4.0 marks
Let [[IMAGE:4cd67eb32c1474f1_4_38]] be an invertible [[IMAGE:4cd67eb32c1474f1_4_39]] -matrix such that [[IMAGE:4cd67eb32c1474f1_4_40]] . Find [[IMAGE:4cd67eb32c1474f1_4_41]] ?
A published solution is not available for this question yet.
Question 7 NAT · 4.0 marks
Let
[[IMAGE:4cd67eb32c1474f1_5_42]]
If [[IMAGE:4cd67eb32c1474f1_5_43]] forms a subspace of [[IMAGE:4cd67eb32c1474f1_5_44]] , then find the dimension of [[IMAGE:4cd67eb32c1474f1_5_45]] . Otherwise, enter the number
[[IMAGE:4cd67eb32c1474f1_5_46]] as the answer.
A published solution is not available for this question yet.
Question 8 NAT · 4.0 marks
Define a [[IMAGE:4cd67eb32c1474f1_5_47]] -matrix [[IMAGE:4cd67eb32c1474f1_5_48]] as [[IMAGE:4cd67eb32c1474f1_5_49]] . Find the rank of the matrix [[IMAGE:4cd67eb32c1474f1_5_50]] .
A published solution is not available for this question yet.
Question 9 NAT · 4.0 marks
Consider the vectors [[IMAGE:4cd67eb32c1474f1_5_51]] and [[IMAGE:4cd67eb32c1474f1_5_52]] in [[IMAGE:4cd67eb32c1474f1_5_53]] , for some [[IMAGE:4cd67eb32c1474f1_5_54]] .
Find the number of values of [[IMAGE:4cd67eb32c1474f1_5_55]] for which the given three vectors are linearly dependent in [[IMAGE:4cd67eb32c1474f1_5_56]] .
A published solution is not available for this question yet.
Question 10 MSQ · 5.0 marks
[[IMAGE:4cd67eb32c1474f1_6_57]] denotes the vector space consisting of real square matrices of order 3 with usual matrix
addition and scalar multiplication. Consider the following matrices:
[[IMAGE:4cd67eb32c1474f1_6_58]]
Use the given information to answer the subquestions.
Which of the following options are correct?
[[IMAGE:4cd67eb32c1474f1_6_59]] , where [[IMAGE:4cd67eb32c1474f1_6_60]] denotes the zero matrix of order 3.
[[IMAGE:4cd67eb32c1474f1_6_61]] , where [[IMAGE:4cd67eb32c1474f1_6_62]] denotes the zero matrix of order 3.
[[IMAGE:4cd67eb32c1474f1_6_63]] , where [[IMAGE:4cd67eb32c1474f1_6_64]] denotes the zero matrix of order 3.
The pair [[IMAGE:4cd67eb32c1474f1_6_65]] is unique for which [[IMAGE:4cd67eb32c1474f1_6_66]] holds, where [[IMAGE:4cd67eb32c1474f1_6_67]] denotes
the zero matrix of order 3.
There is more than one pair [[IMAGE:4cd67eb32c1474f1_6_68]] such that [[IMAGE:4cd67eb32c1474f1_6_69]] .
A published solution is not available for this question yet.
Question 11 MCQ · 4.0 marks
[[IMAGE:4cd67eb32c1474f1_6_57]] denotes the vector space consisting of real square matrices of order 3 with usual matrix
addition and scalar multiplication. Consider the following matrices:
[[IMAGE:4cd67eb32c1474f1_6_58]]
Use the given information to answer the subquestions.
Which of the following options is true?
[[IMAGE:4cd67eb32c1474f1_7_70]] is a linearly independent set.
[[IMAGE:4cd67eb32c1474f1_7_71]] is a linearly independent set.
[[IMAGE:4cd67eb32c1474f1_7_72]] is a linearly dependent set.
[[IMAGE:4cd67eb32c1474f1_7_73]] is a linearly dependent set.
A published solution is not available for this question yet.
Question 12 MSQ · 5.0 marks
Let [[IMAGE:4cd67eb32c1474f1_7_74]] be a coefficient matrix, and [[IMAGE:4cd67eb32c1474f1_7_75]] be a column matrix to a system of [[IMAGE:4cd67eb32c1474f1_7_76]] linear equations with [[IMAGE:4cd67eb32c1474f1_7_77]]
variables, [[IMAGE:4cd67eb32c1474f1_7_78]] , given by [[IMAGE:4cd67eb32c1474f1_7_79]]
where [[IMAGE:4cd67eb32c1474f1_7_80]] .
Use this information to answer the given subquestions.
Let [[IMAGE:4cd67eb32c1474f1_7_81]] denotes the column space of [[IMAGE:4cd67eb32c1474f1_7_82]] . Choose the correct options.
If [[IMAGE:4cd67eb32c1474f1_7_83]] , then there exists vector [[IMAGE:4cd67eb32c1474f1_7_84]] satisfying [[IMAGE:4cd67eb32c1474f1_7_85]] .
If [[IMAGE:4cd67eb32c1474f1_7_86]] , then there exists a unique vector [[IMAGE:4cd67eb32c1474f1_7_87]] satisfying [[IMAGE:4cd67eb32c1474f1_7_88]] .
If there exists a vector [[IMAGE:4cd67eb32c1474f1_7_89]] satisfying the [[IMAGE:4cd67eb32c1474f1_7_90]] , then [[IMAGE:4cd67eb32c1474f1_7_91]] .
If [[IMAGE:4cd67eb32c1474f1_7_92]] , then the system of linear equations [[IMAGE:4cd67eb32c1474f1_7_93]] is consistent.
If [[IMAGE:4cd67eb32c1474f1_7_94]] , then the system of linear equations [[IMAGE:4cd67eb32c1474f1_7_95]] is inconsistent.
A published solution is not available for this question yet.
Question 13 NAT · 4.0 marks
Let [[IMAGE:4cd67eb32c1474f1_7_74]] be a coefficient matrix, and [[IMAGE:4cd67eb32c1474f1_7_75]] be a column matrix to a system of [[IMAGE:4cd67eb32c1474f1_7_76]] linear equations with [[IMAGE:4cd67eb32c1474f1_7_77]]
variables, [[IMAGE:4cd67eb32c1474f1_7_78]] , given by [[IMAGE:4cd67eb32c1474f1_7_79]]
where [[IMAGE:4cd67eb32c1474f1_7_80]] .
Use this information to answer the given subquestions.
[[IMAGE:4cd67eb32c1474f1_8_96]]
[[IMAGE:4cd67eb32c1474f1_8_97]]
Let and . Find the value of [[IMAGE:4cd67eb32c1474f1_8_98]] , up to two decimal places, such that the system
of linear equations [[IMAGE:4cd67eb32c1474f1_8_99]] does not have any solution.
A published solution is not available for this question yet.