Let $A$ and $B$ be two $n×n$ matrices over real numbers. Let rank($M$)…

GATE · 2020 · CS · Computer Science & IT

Let $A$ and $B$ be two $n×n$ matrices over real numbers. Let rank( $M$ ) and det( $M$ ) denote the rank and determinant of a matrix $M$ , respectively. Consider the following statements.

I. rank( $AB$ ) = rank( $A$ )*rank ( $B$ ) II. det( $AB$ ) = det( $A$ )*det( $B$ ) III. rank( $A+B$ ) ≤ rank( $A$ ) + rank( $B$ ) IV. det( $A+B$ ) ≤ det( $A$ ) + det( $B$ )

Which of the above statements are TRUE ?

A.
I and II only
B.
I and IV only
C.
II and III only
D.
III and IV only

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Let \(A\) and \(B\) be two \(n×n\) matrices over real numbers. Let rank(\(M\))…