GATE CS 2020 Set 1 — Question 37
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Engineering Mathematics → Linear Algebra → Matrices & Determinants
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Question
Let and be two matrices over real numbers. Let and denote the rank and determinant of a matrix , respectively. Consider the following statements.I.
II.
III.
IV.
Which of the above statements are TRUE?
II.
III.
IV.
Which of the above statements are TRUE?
Correct answer
(C) II and III only
Solution
Let's evaluate each statement:I.
This is FALSE. The correct property is . For example, if and , , but .II.
This is TRUE. The determinant of a product is the product of the determinants.III.
This is TRUE. This is the subadditivity property of rank.IV.
This is FALSE. For example, let and . . , so . But . .Thus, statements II and III are true.
This is FALSE. The correct property is . For example, if and , , but .II.
This is TRUE. The determinant of a product is the product of the determinants.III.
This is TRUE. This is the subadditivity property of rank.IV.
This is FALSE. For example, let and . . , so . But . .Thus, statements II and III are true.
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