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101.

If space straight A equals open square brackets table row 2 cell space minus 1 end cell row 3 cell space space space space 1 end cell end table close square brackets space comma space straight B equals space open square brackets table row 1 cell space space 3 end cell row 2 cell space 1 end cell end table close square brackets

verify that (A + B) (A – B) ≠ A² – B².

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102. $If space straight A equals open square brackets table row 0 cell space space space 0 end cell cell space space 1 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 1 cell space space 0 end cell cell space space 0 end cell end table close square brackets space comma$ find A². Hence find the matrix A⁶.

Here space straight A equals open square brackets table row 0 cell space space 0 end cell cell space space 1 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 1 cell space space 0 end cell cell space space 0 end cell end table close square brackets therefore space space straight A squared equals space AA equals open square brackets table row 0 cell space space 0 end cell cell space space 1 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 1 cell space space 0 end cell cell space space 0 end cell end table close square brackets open square brackets table row 0 cell space space 0 end cell cell space space 1 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 1 cell space space 0 end cell cell space space 0 end cell end table close square brackets space space space space equals open square brackets table row cell 0 plus 0 plus 1 end cell cell space space 0 plus 0 plus 0 end cell cell space space 0 plus 0 plus 0 end cell row cell 0 plus 0 plus 0 end cell cell space space 0 plus 1 plus 0 end cell cell space space 0 plus 0 plus 0 end cell row cell 0 plus 0 plus 0 end cell cell space space 0 plus 0 plus 0 end cell cell space space 1 plus 0 plus 0 end cell end table close square brackets space equals open square brackets table row 0 cell space space 0 end cell cell space space 1 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 1 cell space space 0 end cell cell space space 0 end cell end table close square brackets equals 1 NOW space straight A to the power of 6 space equals left parenthesis straight A squared right parenthesis cubed space equals left parenthesis 1 right parenthesis cubed equals 1 therefore space straight A to the power of 6 space equals open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets

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103.

IF space matrix space straight A equals open square brackets table row 0 cell space 0 end cell row 4 cell space 0 end cell end table close square brackets

,find A¹⁶

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104.

if space open square brackets table row 1 0 row 0 1 end table close square brackets comma space straight A equals open square brackets table row 0 1 row 0 0 end table close square brackets comma space then space show space that left parenthesis straight a space straight I space plus straight b space straight A right parenthesis cubed equals straight a cubed straight I plus 3 space straight a squared space bA.

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105. Solve for x and y, given that

open square brackets table row 3 cell space space minus 4 end cell row 1 cell space space space space space 2 end cell end table close square brackets space open square brackets table row x row y end table close square brackets equals open square brackets table row cell space 3 end cell row 11 end table close square brackets

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106. Solve for.x and y, given that

open square brackets table row x y row cell 3 y end cell x end table close square brackets open square brackets table row 1 row 2 end table close square brackets equals open square brackets table row 3 row 5 end table close square brackets

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107.

Solve space for space straight x space andy comma space given space that open square brackets table row 2 cell negative 3 end cell row 1 1 end table close square brackets space open square brackets table row straight x row straight y end table close square brackets equals open square brackets table row 1 row 3 end table close square brackets

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108.

Find space the space values space of space straight a space and space straight b space for space which space the space following space holds space colon open square brackets table row straight a straight b row cell negative straight a end cell cell 2 straight b end cell end table close square brackets space open square brackets table row 2 row cell negative 1 end cell end table close square brackets equals open square brackets table row 5 row 4 end table close square brackets

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109.

Find space the space values space of space straight a space and space straight b space for space which space the space following space holds space colon open square brackets table row 4 2 row 3 cell negative 1 end cell end table close square brackets space open square brackets table row straight a row straight b end table close square brackets equals space open square brackets table row cell negative 4 end cell row 2 end table close square brackets

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110.

Let space straight A equals open square brackets table row 2 cell negative 1 end cell row 3 4 end table close square brackets straight B equals space open square brackets table row 5 2 row 7 4 end table close square brackets comma space straight C equals space open square brackets table row 2 5 row 3 8 end table close square brackets space Find space straight a space matrix space straight D space such space that space CD minus AB equals 0

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102. find A2. Hence find the matrix A6.