If matrix AB = O , then

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3 and 4 .Determinants and Matrices

normal

If the matrix $AB = O$, then

$A = O$ or $B = O$

$A = O$ and $B = O$

It is not necessary that either $A = O$or $B = O$

$A \ne O,B \ne O$

Solution

==> $|A|\,.\,|B| = 0$ ==> $|A|\, = \,0$ or $|B|\, = \,0$

When $AB = O$, neither $A$ nor $B$ may be $O.$

For example if $A = \left[ {\begin{array}{*{20}{c}}1&0\\0&0\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}0&0\\1&0\end{array}} \right]$, then $AB = \left[ {\begin{array}{*{20}{c}}0&0\\0&0\end{array}} \right]$.

Standard 12

Mathematics

$A$ is a $2 × 2$ matrix such that $A$ $\left[ {\begin{array}{{20}{c}}1\\{ – 1} \end{array}} \right]$ $=$$\left[ {\begin{array}{{20}{c}}{ – 1}\\2\end{array}} \right]$ and $A^2$ $\left[ {\begin{array}{{20}{c}}1\\{ – 1}\end{array}} \right]$ $=$ $\left[ {\begin{array}{{20}{c}}1\\0\end{array}} \right]$ . The sum of the elements of $A$, is

normal

Let $A$ be a square matrix such that ${a_{ij}} \in \left\{ { – 1,0,1} \right\}\forall\,\, i,j$ and it has only one non-zero entry in each row as well as in each column, then

hard

The bookshop of a particular school has $10 $ dozen chemistry books, $8$ dozen physics books, $10$ dozen economics books. Their selling prices are Rs. $80,$ Rs. $60$ and Rs. $40$ each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

medium

If $\left[ {\begin{array}{{20}{c}}{x + y}&{2x + z}\\{x – y}&{2z + w}\end{array}} \right] = \left[ {\begin{array}{{20}{c}}4&7\\0&{10}\end{array}} \right]$, then values of $x, y, z, w$ are

easy

If $A = \left[ {\begin{array}{*{20}{c}}0&i\\{ – i}&0\end{array}} \right]$, then the value of ${A^{40}}$ is