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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.

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

Here space straight I space equals space open square brackets table row 1 0 row 0 1 end table close square brackets space space space space space space straight A equals space open square brackets table row 0 1 row 0 0 end table close square brackets therefore space straight a space straight I equals space open square brackets table row straight a 0 row 0 straight a end table close square brackets space comma space bA space equals space open square brackets table row 0 straight b row 0 straight a end table close square brackets therefore space straight a space straight I space plus straight b space straight A space equals space open square brackets table row straight a straight b row 0 straight a end table close square brackets therefore space left parenthesis straight a space straight I space plus space straight b space straight A right parenthesis squared equals space open square brackets table row straight a straight b row 0 straight a end table close square brackets space space open square brackets table row straight a straight b row 0 straight a end table close square brackets space equals space open square brackets table row cell straight a squared end cell cell 2 ab end cell row 0 cell straight a squared end cell end table close square brackets space space space space space space space left parenthesis straight a space straight I space plus space straight b space straight A right parenthesis cubed equals space open square brackets table row cell straight a squared end cell cell 2 ab end cell row 0 cell straight a squared end cell end table close square brackets space open square brackets table row straight a straight b row 0 straight a end table close square brackets space equals space open square brackets table row cell straight a cubed end cell cell 3 straight a squared straight b end cell row 0 cell straight a cubed end cell end table close square brackets therefore space space space left parenthesis straight a space straight I space plus space straight b space straight A right parenthesis cubed equals space space space open square brackets table row cell straight a cubed end cell cell 3 straight a squared straight b end cell row 0 cell straight a cubed end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Also space space space space space straight a to the power of 3 space space end exponent straight I space plus 3 straight a squared space straight b space straight A space equals space straight a cubed space open square brackets table row 1 0 row 0 1 end table close square brackets plus 3 straight a squared space straight b space open square brackets table row 0 1 row 0 0 end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open square brackets table row cell straight a cubed end cell cell 3 straight a to the power of 2 space end exponent straight b end cell row 0 cell straight a cubed end cell end table close square brackets space plus open square brackets table row 0 cell 3 straight a squared straight b end cell row 0 cell straight a cubed end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space therefore space space space space space space space space straight a to the power of 3 space end exponent space straight I equals 3 straight a to the power of 2 space end exponent straight b space straight A equals space open square brackets table row cell straight a cubed end cell cell 3 straight a to the power of 2 space end exponent straight b end cell row 0 cell straight a cubed end cell end table close square brackets space space space space space space space space space From space left parenthesis 1 right parenthesis space space and space left parenthesis 2 right parenthesis. space we space get comma left parenthesis straight a space straight I space plus straight b space straight A space right parenthesis cubed space space space space space space space equals straight a cubed space space space straight I space plus 3 straight a squared space 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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