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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.

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

space straight A equals open square brackets table row 2 cell negative 1 end cell row 3 4 end table close square brackets comma straight B equals open square brackets table row 5 2 row 7 4 end table close square brackets comma straight C equals open square brackets table row 2 5 row 3 8 end table close square brackets Let straight D equals open square brackets table row straight x straight y row straight z straight t end table close square brackets Now space CD minus AB equals 0 rightwards double arrow CD equals AB rightwards double arrow space space open square brackets table row 2 5 row 3 8 end table close square brackets space open square brackets table row straight x straight y row straight z straight t end table close square brackets space equals space open square brackets table row 2 cell negative 1 end cell row 3 4 end table close square brackets space open square brackets table row 5 2 row 7 4 end table close square brackets rightwards double arrow open square brackets table row cell 2 straight x plus 5 yz end cell cell 2 straight y plus 5 straight t end cell row cell 3 straight x plus 8 end cell cell 3 straight y plus 8 straight t end cell end table close square brackets equals open square brackets table row cell 10 minus 7 end cell cell 4 minus 4 end cell row cell negative 15 plus 28 end cell cell 6 plus 16 end cell end table close square brackets rightwards double arrow open square brackets table row cell 2 straight x plus 5 yz end cell cell 2 straight y plus 5 straight t end cell row cell 3 straight x plus 8 end cell cell 3 straight y plus 8 straight t end cell end table close square brackets equals space open square brackets table row 3 0 row 43 22 end table close square brackets By space definition space of space equality space of space matrices comma space space 2 straight x space plus space 5 space straight z space equals space 3 space space space space space space space space.. left parenthesis 1 right parenthesis space space 3 straight x space plus space 8 space straight z space equals space 43 space space space space.. left parenthesis 2 right parenthesis space space 2 straight y space plus space 5 space straight t space equals space 0 space... left parenthesis 3 right parenthesis space space 3 straight y space plus space 8 space straight t equals 22 space... left parenthesis 4 right parenthesis space space Multiplying space left parenthesis 1 right parenthesis space by space 8 space and space left parenthesis 2 right parenthesis space by space 5 comma space we space get comma space space 16 space straight x space plus space 40 space straight z space equals space 24 space space space space.. left parenthesis 5 right parenthesis space space 15 semicolon straight x space plus space 40 straight z space equals space 215 space space space space... left parenthesis 6 right parenthesis space space Subtracting space left parenthesis 6 right parenthesis space from space left parenthesis 5 right parenthesis comma space we space get comma space space straight x space equals space – space 191 space space therefore space from left parenthesis 1 right parenthesis comma space – space 382 space plus space 5 space straight z space equals space 3 space equals greater than space 5 straight z space equals space 385 space rightwards double arrow space straight z space equals space 77 space space Multiplying space left parenthesis 3 right parenthesis space by space 8 space and space left parenthesis 4 right parenthesis space by space 5 comma space we space get comma space space 16 straight y space plus space 40 space straight t space space equals space 0 space space space space... left parenthesis 7 right parenthesis 15 straight y space plus space 40 space straight t space equals space 110 space space space space... left parenthesis 8 right parenthesis space space Subtracting space left parenthesis 8 right parenthesis space from space left parenthesis 7 right parenthesis comma space we space get comma space straight y space equals space – space 110 space space therefore space from space left parenthesis 3 right parenthesis comma space – 220 space plus space 5 space straight t space equals space 0 space rightwards double arrow space 5 straight t space equals space 220 space rightwards double arrow straight t space equals space 44 space space therefore we space have straight x space equals space – space 191 comma space straight y space equals space – space 110 comma space straight z space equals space 77 comma space straight t space equals space 44 therefore space straight D 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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