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  • Give answer! If A and B are two non–zero square matrices of the same order n such that AB = O, then |A| = |B| = 0. Then which of the following is true

If A and B are two non–zero square matrices of the same order n such that AB = O, then |A| = |B| = 0. Then which of the following is true

  • Option 1)

    \left | C \right |=0\Leftrightarrow \left | adjC \right |=1

  • Option 2)

    \left | C \right |=0\Leftrightarrow \left | adjC \right |=0

  • Option 3)

    \left | C \right |=1\Leftrightarrow \left | adjC \right |=0

  • Option 4)

    none of these 

 

Answers (1)

best_answer

  As we have learned

Solution of a system of equations -

x_{1},x_{2},\cdot \cdot \cdot ,x_{n} satisfy the system of linear equations  Ax=B

- wherein

 

 

If |A| \neq 0, then A–1 exists. Hence  AB = 0 \Rightarrow A–1 (AB) = A–1O \Rightarrow B = O, which is not the case. Hence |A| = 0. Similarly |B| = 0.

            If C = O, then obviously adjC = O and hence the result is obviously true. Now consider the case that C \neq O. We know that C(adj C) = |C| I. If |C| = 0, then C(adj C) = O. If |adjC| \neq 0, then as earlier, it can be shown that C = 0, which is not the case. Hence |adj C| = 0.

            For the converse if adjC = O, then obviously |C| = 0.

            If adj C \neq 0 (given that |adj C| = 0), then C(adj C) = |C|I \Rightarrow  |C|  |adj C| = |C|n. If |C| \neq 0 then |adj C| \neq 0, which is not the case. Hence |C| = 0

 


Option 1)

\left | C \right |=0\Leftrightarrow \left | adjC \right |=1

Option 2)

\left | C \right |=0\Leftrightarrow \left | adjC \right |=0

Option 3)

\left | C \right |=1\Leftrightarrow \left | adjC \right |=0

Option 4)

none of these 

Posted by

gaurav

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