Let $straight A space equals space open parentheses table row 2 cell negative 1 end cell row 3 4 end table close parentheses comma space straight B space equals space open parentheses table row 5 2 row 7 4 end table close parentheses comma space straight C equals space open parentheses table row 2 5 row 3 8 end table close parentheses$ find a matrix D such that CD – AB = O.

Solution

Let space straight D space equals space open square brackets table row straight p straight q row straight r straight s end table close square brackets

Then CD-AB = O

open square brackets table row 2 5 row 3 8 end table close square brackets open square brackets table row straight p straight q row straight r straight s end table close square brackets space minus space open square brackets table row 2 cell negative 1 end cell row 3 cell space space 4 end cell end table close square brackets open square brackets table row 5 2 row 7 4 end table close square brackets space equals straight O rightwards double arrow space open square brackets table row cell 2 straight p space plus space 5 straight r end cell cell 2 straight q space plus space 5 straight s end cell row cell 3 straight p space plus space 8 straight r end cell cell 3 straight q space plus space 8 straight s end cell end table close square brackets minus open square brackets table row 3 0 row 43 22 end table close square brackets space equals space open square brackets table row 0 0 row 0 0 end table close square brackets rightwards double arrow space open square brackets table row cell 2 straight p plus space 5 straight r minus 3 end cell cell 2 straight q space plus 5 straight s end cell row cell 3 straight p plus 8 straight r minus 43 end cell cell 3 straight q plus 8 straight s minus 22 end cell end table close square brackets space equals space open square brackets table row 0 0 row 0 0 end table close square brackets

By equality of matrices we get,
2p +5r-3 = 0 ....(1)
3p +8r-43 = 0 ..(2)
2q +5s = 0 ......(3)
3q +8s-22 = 0 ..(4)
By solving (1) and (2) we get p = -191 and r = 77
Similarly, on solving (3) and (4) we get q = - 110 and s = 44
$Let space straight D space equals space open square brackets table row straight p straight q row straight r straight s end table close square brackets space equals space open square brackets table row cell negative 191 end cell cell negative 110 end cell row 77 44 end table close square brackets$

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Matrices

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